How was Parushurama's pride (ego) destroyed?

It is said that Parashurama was the sixth incarnation of Lord Vishnu and is seen in both Ramayana and Mahabharata. He is well known for his pride as well as valor. We also know that he gave the the great bow called Shiva dhanush to Raja Janak, the father of Sita and we also know that Lord Rama broke this bow and married Sita.

After the marriage everyone were really happy that the marriage ceremonies went on well. Everyone including Sita and her sisters Urmila, Mandavi and Shruthakeerthi, who were married to Lakshmana, Bharata and Shatrugna respectively were ready to go to Ayodya. 

Suddenly from nowhere, appeared the great Parashuram. Everyone were really scared as they know about his angry nature too. He came to Rama and said "Where are you heading towards O! Ram. I heard that you broke my bow and married Janaki. That's a very old bow which was rusted and lying aside for many years. Now, take this challenge and if possible hold this bow." and showed him the Vishnu Dhanush. Everyone were taken aback. Immediately Dhasharath Maharaj came forward and pleaded Parashuram that Lord Ram is a small boy and that would be a gargantuan task for him to handle. Lord Parashuram brushed him and wanted an answer from only Lord Rama. 

Lord Rama in turn was very calm and poised as usual and spoke " Oh great Maharishi! I am ready for the challenge, but if I tie the thread in the bow then I need to use the arrow and if use the arrow then I need to shoot at something. So tell me what should I shoot at? If I shoot at such a great Maharishi like you, then it would be sin. So make it clear for me." 

Lord Parashuram started laughing at Lord Rama and told him to first lift the bow and then think about the target. But Lakshmana became very angry and started supporting his beloved brother and insisted Lord Parashuram to fix the target. Lord Parashuaram accepted and told him to have all the boons which he has got through his penance as the target.

Lord Rama accepted the challenge and lifted the Vishnu Dhanush in one hand as he did the Shiva Dhanush and tied the thread. Then, he took an arrow and as said by Lord Parashuaram shot an arrow at the target. Lord Parashuram understood who it was and bowed to Lord Rama. Finally he realised that he was the incarnation of Lord Vishnu and saw at Rishi vashisht who also shook his head. He finally realised his ego was crushed and without speaking a word turned and went back to do penance.