Why is Kylo Ren fixated on just one of his parents?

Question

Kylo Ren mentions or alludes to his father multiple times in the movie, and in the end,

murders his father in cold blood

as some sort of "proving his worth" or I dunno what.

He never once mentions his mother,

even though she's the one who is related to his idolized grandfather. She's also more important politically speaking: his dad is just an old recidivist smuggler, while his mom is a freakin' General. It seems like Snoke could've killed two birds with one stone if he had urged his disciple to kill Mom, but instead, he, too only ever mentions Han, not Leia.

Why the daddy-fixation?

Answer 1

Kylo believes that Han was a disappointing father. At the same time, ideological male figures dominate his world view.

After Kylo scans Rey's mind and finds that she considers Han to be like a "father she never had", he tells her,

"He would have disappointed you."

We have no evidence of Kylo's opinion of Leia as you point out, but he clearly believes that Han's parenting was quite deficient, enough to fixate on it and vocalize it.

Also, Kylo's world view is filled with male figures seducing each other from one side to another: Vader attempting to seduce Luke to the Dark Side, Luke attempting (and succeeding) to seduce Vader to the Light, Luke trying to train Kylo in the Light, and Snoke seducing Kylo to the Dark. In his eyes, philosophies and fathers are almost equivalent — to be followed or to be rejected.

Answer 2

In-universe

1. Because his Master, Snoke, basically tells him to.

   “Your self-belief is commendable, Kylo Ren, but do not let it blind you. No one knows the limits of his own power until it has been tested to the utmost, as yours has not been. That day may yet come. ...

   ...

   “Perhaps,” Snoke conceded. “It has come to our notice that the droid we seek is aboard the Millennium Falcon, once again in the hands of your father, Han Solo. Even you, master of the Knights of Ren, have never faced such a test.”
   Ren considered his reply carefully. “It does not matter. He means nothing to me. My allegiance is with you. No one will stand in our way.”
   Snoke nodded. “We shall see. We shall see.”

   Basically, he dares Ren to kill his father, both to prove his loyalty to him, AND to the Dark Side.

   (all quotes from Foster's TFA novelization)

2. Because as part of that "telling", father/son dynamics between Vader/Anakin and Luke is important as far as killing/not killing each other. And Kylo is all about his connections to Darth Vader.

   Kylo Ren, I watched the Galactic Empire rise, and then fall. The gullible prattle on about the triumph of truth and justice, of individualism and free will. As if such things were solid and real instead of simple subjective judgments. The historians have it all wrong. It was neither poor strategy nor arrogance that brought down the Empire. You know too well what did.”
   Ren nodded once. “Sentiment.”
   “Yes. Such a simple thing. Such a foolish error of judgment. A momentary lapse in an otherwise exemplary life. Had Lord Vader not succumbed to emotion at the crucial moment — had the father killed the son — the Empire would have prevailed. And there would be no threat of Skywalker’s return today.”

   So Kylo Ren is trying to undo Vader's error and "finish what he started", in reverse generational order.

Out of universe

The writers most likely were making at least some level of allusion to an Oedipus complex (which is of course named after a character in Sophocles who kills his father). Predictably, since so were the writers of Original Trilogy.

I shall leave any FURTHER parallels to the story of Skywalkers and Leia unsaid, for my own sanity, and leave them as an exercise to the reader fanfic writers.

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As a side note, i'm not sure your theory that Kylo Ren is "focused on just one parent" is even right - as opposed to Han simply being an earlier opportunity.

Note this snippet of their conversation:

“I am immune to the light,” Ren assured him confidently. “By the grace of your training, I will not be seduced.”
“Your self-belief is commendable, Kylo Ren, but do not let it blind you. No one knows the limits of his own power until it has been tested to the utmost, as yours has not been. That day may yet come. There has been an awakening in the Force. Have you felt it?”
Ren nodded. “Yes.”
“The elements align, Kylo Ren. You alone are caught in the winds of the storm. Your bond is not just to Vader, but to Skywalker himself. Leia…”
“There is no need for concern.” Despite the Supreme Leader’s cautioning, Ren’s assurance remained unbounded. “Together we will destroy the Resistance—and the last Jedi.”

As you can see, they both mention Leia, and intend to destroy her as part of a Resistance.

Answer 3

I do not think that there is any evidence yet that Kylo Ren is focussed on only of his parents. Sure, he might very well be, but he might as well be fixated on both his parents equally.

Leaving aside what can be deduced from Leia’s leading position in the Resistance, there is no indication that Kylo Ren knows of Leia’s involvement in the events of Episode VII. By contrast, he is reminded of and confronted with Han Solo directly several times:

• He learns that the droid he is looking for is accompanied by Han Solo (quote from DVK’s answer):

  “It has come to our notice that the droid we seek is aboard the Millennium Falcon, once again in the hands of your father, Han Solo. Even you, master of the Knights of Ren, have never faced such a test.”

  This is in fact the point at which the viewers learn about Kylo’s lineage in the first place. Thus there couldn’t be much opportunity or trigger for Kylo to express his attitude towards his parents in general before this, and afterwards all his words and actions concerning his father could be triggered by this and not by personal preference.

• By mindreading her, Kylo learns of Rey’s connection to Han. She has never met Leia at this point.

• Han directly confronts him on Starkiller Base, and Kylo murders him as a result.

It is through these events that we get to know Kylo’s attitude towards his father, and I find it very well conceivable that Kylo talks only about his father because it was him and not his mother who was brought to his attention and eventually prancing in front of him.

Answer 4

One thing to add which I'm surprised wasn't in the above answers - the film shows him going from disdainful indifference to being tasked to kill his father, while he was already at war with his mother and had been for years. Leia was already his nemesis and (from his perspective) mortal rival.

Presumably, he already went through the same mental turmoil with Leia years earlier. He's not mulling over the decision because he made it years ago. He's not disparaging Leia like he does Han now, because he doesn't need to re-justify killing her to himself, he's already invested in that decision, her being his enemies' general makes it pretty easy to continue to justify it to himself, and has been actively and constantly at war with her ever since. It's the status quo.

With his father, however, he's had no reason to even think about him for years. He needs to work himself up to patricide from almost nothing.

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From Snoke's point of view (the closely related question why does Snoke push for patricide and not matricide which would be militarily more advantageous), as well as the fact that motivating Kylo to want to commit matricide is a battle he appears to have won long ago, force types can sometimes sense major events coming. Snoke sensed a fateful Kylo-Han interaction, and pushed him to prepare for it. No fateful Kylo-Leia interaction appeared to be on the cards, however, so it simply wasn't as topical.

Answer 5

My answer - which I will explain further in my post - is that it is entirely likely that the topic of Leia is perhaps an even more painful subject matter for Kylo Ren to think about much less actually talk about... which is why he appears distant and far removed from her.

Though Kylo mentions little of anything of his mother in the movie, in the book, he reacts to her being mentioned by his own father... and with significant meaning.

“No, it’s not.” Halfway across the walkway now, Han continued to move forward, smiling. “Never too late for the truth. Leave here with me. Come home.” Without the slightest trace of malice or deception, he cast a dagger. “Your mother misses you.”

A strange sensation touched the younger man’s cheeks. Something long forgotten. Dampness. Tears.

“I’m being torn apart. I want—I want to be free of this pain.”

With the emphasis being made on the word "dagger" because it's explicitly being applied to, "Your mother misses you," and coupled with the description of, "long forgotten," the conclusion that Kylo Ren is distant from his mother can be reliably drawn...

However, at the same time:

Why describe reminding Kylo Ren about his mother as a "dagger"?

Why is Han's mention of said mother so specifically described as lacking both "malice" and "deception"?

My answer to the whys - which is my answer - is that far from being an unimportant figure to Kylo Ren, the topic of Leia/his mother is actually a particularly painful subject...

to the point that he's not only attempted to forget about her, but also if it wasn't Han/Leia's former? husband/Kylo Ren's father who was mentioning Leia to Kylo Ren, then the mention of Leia by anyone else would be considered to be related to an attempt at "malice" and "deception"...

or, in other words, a deliberate attempt at being cruel towards him (Kylo Ren) or an attempt to try and somehow trick him.

To further support this, regarding @DVK's quoted bit of conversation, Kylo Ren interrupts Snoke before anything more can be said regarding Leia.

And lastly, as to why Leia is likely a particularly painful subject... it's because Kylo Ren is - and had been - "torn apart"... and for what was/is likely a very long time.

Leia is 'Light'. Kylo Ren is 'Dark'.

Unlike Han who has become a smuggler again and who has, throughout canonical history, shown himself to have both 'Light' and more 'Dark' moments, Leia has canonically always been shown as a member of the 'Light'.

Add on the fact that Leia tells Han that she knew of Snoke's involvement (but didn't want to involve Han):

She sighed. “Many reasons. I was hoping that I was wrong, that it wasn’t true. I hoped I could sway him, turn him away from the dark side, without having to involve you.” A small smile appeared. “You had— you have— wonderful qualities, Han, but patience and understanding were never among them. I was afraid that your reactions would only drive him farther to the dark side. I thought I could shield him from Snoke’s influence and you from what was happening.”

... cut for brevity ...

“Always,” she told him. “From the shadows, in the beginning, even before I realized what was happening, he was manipulating everything, pulling our son toward the dark side. But nothing’s impossible, Han. Not even now, at this late time. I have this feeling that if anyone can save him— it’s you.”

And it becomes entirely likely that the 'Light' versus 'Dark' battle with Kylo Ren started since near the beginning which would have put him in a position of great conflict with Leia/his mother starting at the same time as she - because she never told Han - was the primary 'disciplinarian' figure.

Snoke tried pulling, Leia tried pulling back, fighting back, cajoling, whatever.

Whatever it was she did, Kylo Ren was being pulled one way by Snoke and being pulled in another way by Leia/his mother who was the primary disciplinarian figure and it was something that contributed to his "torn" state that he tells to Han.

Though Darth Vader is indeed Kylo Ren's grandfather by virtue of Leia, Darth Vader - not Anakin Skywalker who became Vader and who was later redeemed - and Leia were also complete opposites in terms of Force-leanings with Leia canonically never budging an inch from her 'Light' side leanings and with Darth Vader being a case of Light-gone-Dark and last minute redemption.