No. He never loved anyone.
JK Rowling confirmed this in a Q&A in 2004.
Has Voldemort or Tom Riddle ever cared for or loved anyone?
Now, that’s a cracking question to end with—very good. No, never. If he had, he couldn’t possibly be what he is.
(JK Rowling at the Edinburgh Book Festival, Accio Quote).
As for why he killed his father, his motive was pure Muggle-hatred. As a rabid blood purist Voldemort's family was an embarrassment to him. His surname was a constant reminder of his Muggle lineage through the Riddle line. Killing his father and two grandparents was his way of removing that part of his history and identity. He saw their deaths as a pruning of his own family tree.
“You stand, Harry Potter, upon the remains of my late father,” he hissed softly. “A Muggle and a fool...very like your dear mother. But they both had their uses, did they not? Your mother died to defend you as a child...and I killed my father, and see how useful he has proved himself, in death...”
Voldemort laughed again. Up and down he paced, looking all around him as he walked, and the snake continued to circle in the grass.
“You see that house upon the hillside, Potter? My father lived there. My mother, a witch who lived here in this village, fell in love with him. But he abandoned her when she told him what she was...He didn’t like magic, my father...
“He left her and returned to his Muggle parents before I was even born, Potter, and she died giving birth to me, leaving me to be raised in a Muggle orphanage...but I vowed to find him...I revenged myself upon him, that fool who gave me his name...Tom Riddle.
(Goblet of Fire, Chapter 33, The Death Eaters).
Would Voldemort have felt bitter towards his father for abandoning his mother? Quite possibly. But the reason that he killed him was because he wasn't a wizard. Voldemort viewed him as "a Muggle and a fool". By killing his father Voldemort believed that he was revenging himself for being related to a Muggle, not revenging his mother in any way. The premise in the question is correct in this regard.
On the only occasion we hear Voldemort talk about his mother he criticises her for her mortality. This can hardly be described as love.
“My mother can’t have been magic, or she wouldn’t have died,” said Riddle, more to himself than Dumbledore.
(Half-Blood Prince, Chapter 13, The Secret Riddle).