I have heard different versions of this story, especially Parashurama's response. Hence, this question. I would like to know the original story mentioned in Valmiki's Ramayana.
This question discusses why they engaged in a duel (assuming that they did) but does not talk about the story of the assumed duel itself, which I am interested in.
Also, this question talks about the reasons why Shri Rama was able to lift the Dhanush, without talking about the original story itself.
To answer the first part of the question
Did Shri Rama break Parashurama's Dhanusha during the Swayamvara?
According to Valmiki's Ramayana, that bow was of Lord Shiva and he gave it to Devaraata.
देवरात इति ख्यातो निमेः ज्येष्ठो मही पतिः |
न्यासो अयम् तस्य भगवन् हस्ते दत्तो महात्मना || १-६६-८Oh, godly sage, there was a king renowned as Devaraata, sixth one from Nimi, [the originator of our lineage,] and this bow was handed down to him for custodial care by the Sublime Soul, Shiva...
Source: http://www.valmikiramayan.net/utf8/baala/sarga66/bala_66_frame.htm
So, this bow was of Lord Shiva not Lord Parashurama.
And then Lord Rama break that bow easily:
पश्यताम् नृ सहस्राणाम् बहूनाम् रघुनंदनः |
आरोपयत् स धर्मात्मा स लीलम् इव तत् धनुः || १-६७-१६While many thousands of men are witnessing that right-minded Rama the legatee of Raghu stringed the bow effortlessly.
आरोपयित्वा मौर्वीम् च पूरयामास वीर्यवान् |
तत् बभंज धनुर् मध्ये नरश्रेष्ठो महायशाः || १-६७-१७Further, that dextrous one has stringed that bow with bowstring and started to stretch it up to his ear to examine its tautness, but that glorious one who is foremost among men, Rama, broke that bow medially.
In Valimiki Ramayana, the duel between Dasaratha-Rama and Parashu-Rama that you mentioned in your question does not happen during Sita's Swayamvara, but a little later when Parashurama confronts Dasaratha, Rama & others when they are returning to Ayodhya after the wedding.
Also it seems like you are mixing the two bows because there are two bows:
Shiva's bow (Pinaka): Rama breaks this bow during Sita's Swayamvara. Although Parashurama learns about it, he's nowhere in the vicinity when this happens. Shiva's bow was in Janaka's possession since long time. You can read how Janaka inherited it here.
Vishnu's bow (Sharanga): Parashurama inherits this bow from his father, Sage Jamadagni. When he confronts Rama, Parashurama hands over this bow to Rama inciting him to demonstrate his power. After showing his prowess, Rama doesn't break the bow though, he simply hands it over to Varuna. And Parashurama retreats to Mt. Mahendra. The following describes their encounter in little more detail.
Addressing Rama, Parashurama says:
श्रुत्वा तु धनुषो भेदम् ततो अहम् द्रुतम् आगतः || १-७५-२६
तत् एवम् वैष्णवम् राम पितृ पैतामहम् महत् |
क्षत्र धर्मम् पुरस् कृत्य गृह्णीष्व धनुर् उत्तमम् || १-७५-२७On hearing about the breakage of Shiva's longbow, then I promptly came here. Thereby, oh, Rama, wield this supernatural and superlative longbow of Vishnu, which is passed on to me from my forefathers and my father. Keep your fealty to Kshatriya-hood in view, and wield this as you have wielded Shiva's longbow... [1-75-26b, 27]
योजयस्व धनुः श्रेष्ठे शरम् पर पुरम् जयम् |
यदि शक्तः असि काकुत्स्थ द्वन्द्वम् दास्यामि ते ततः || १-७५-२८Take an aim with an arrow that conquers enemy's citadels fixing it on this supramundane longbow... and oh, Kakutstha, should you be capable of it, thereafter I will give you a duel... [1-75-27]
Rama, after taking Vishnu's bow from Parashurama, replies:
ब्राह्मणो असि इति पूज्यो मे विश्वामित्र कृतेन च |
तस्मात् शक्तो न ते राम मोक्तुम् प्राण हरम् शरम् || १-७६-६Oh, Rama of Bhaargava, even if you are a Brahman you are eliminable, but because of your relationship with Vishvamitra, and because you are a venerable one for me, I am disinclined to release this arrow that exterminates your life... [1-76-6]
इमाम् वा त्वत् गतिम् राम तपो बल समार्जितान् |
लोकान् अप्रतिमान् वा अपि हनिष्यामि यत् इच्छसि || १-७६-७Oh, Bhaargava Rama, either this motility of yours at the speed of your mind, or even those unparalleled realms of heavens which you have earned by the power of your ascesis, I will eliminate whichever you wish... [1-76-7]
Parashurama is overwhelmed by Rama's intensity and tells Rama to aim his arrow toward (and destroy) the regions he (Parashurama) has conquered by his asceticism and spare his course and the Earth:
जडी कृते तदा लोके रामे वर धनुर् धरे |
निर्वीर्यो जामदग्न्यो असौ रमो रामम् उदैक्षत || १-७६-११Then, when Rama is ready to take aim with the arrow on that inscrutable longbow, and when the worlds are being rendered as insentient, then that Rama of Jamadagni is rendered vigourless and he stared up at Rama of Dasharatha. [1-76-11]
तम् इमाम् मत् गतिम् वीर हन्तुम् न अर्हसि राघव |
मनो जवम् गमिष्यामि महेन्द्रम् पर्वत उत्तमम् || १-७६-१५Thereby oh, valiant one, it will be inapt of you to impair this motility of mine, oh, Raghava, I will depart with the speed of cerebration to Mt. Mahendra, a par excellent mountain... [1-76-15]
लोकाः तु अप्रतिमा राम निर्जिताः तपसा मया |
जहि तान् शर मुख्येन मा भूत् कालस्य पर्ययः || १-७६-१६But I triumphed over matchless realms of heavens with my ascesis, oh, Rama, you may hash them up with that irreversible arrow... let there be no time-lag... [1-76-16]
स हतान् दृश्य रामेण स्वान् लोकान् तपसा आर्जितान् |
जामदग्न्यो जगाम आशु महेन्द्रम् पर्वतोत्तमम् || १-७६-२२On seeing all of his realms of heavens are shot-blasted by Rama of Dasharatha, Rama of Jamadagni vanished in a trice to Mt. Mahendra, the heavenly mountain. [1-76-22]
Rama then hands over the Sharanga to Varuna:
गते रामे प्रशांत आत्मा रामो दाशरथिः धनुः |
वरुणाय अप्रमेयाय ददौ हस्ते महायशाः || १-७७-१
On the departure of Rama of Jamadagni, that most glorious Rama of Dasharatha is quietened at heart, and he gave away that longbow of Vishnu into the hand of inimitable Rain-god. [1-77-1]
If you want to read the story of their encounter at a stretch, I found this translation to be much better than the verse by verse translation at valmikiramayan.net.
You may also find these notes from the translators at valmikiramayan.net useful:
Raghava, the nimble-handed vanquisher of his opponents, speaking thus in high dudgeon expropriated that estimable weapon, namely the longbow of Vishnu, from the hand of Bhaargava Rama, along with the long-arrow that is already fitted on it... [1-76-4]
"and, along the same lines Rama also said to have extricated the essential nature of Bhaargava...' The above said 'distancing' of these two Rama-s has another purpose. Rama has certain innate nature of making his observers enchanted by his very personality. Even Ravana looks at him adoringly in the war scene. This apart, his hand has certain ability to extricate the innate nature of others, should he lay his hand on them. We rarely see him touching or patting others, except for Seetha, Lakshmana, Hanuma, and say a squirrel etc. So, it is believed that, Rama has now expropriated the essential nature of Vishnu from Bhaargava Rama, while snatching the bow from his hands.
<p>For this Padma Puraana says: <br></p> <p>iti uktvaa devii vaiShNavyaa shaktyaa tad gataayaa saha | <br> jagraaha vaiShNavam caapam vinayena ca liilayaa || <br></p> <p>'oh, Devi Parvati, <strong>saying so Rama took away the Vishnu's anima from Bhaargava Rama, along with bow of Vishnu, sportily and obediently, too</strong>...' </p> <p>Thus, Rama of Dasharatha bade goodbye to his earlier incarnation, Parashu Rama, as two swords cannot be in one sheath.</p>
