Hari Om Kugan,
Hope you won't think poorly of my just walking in with my thoughts. The Ramayan is always a spource of great joy, hence here some thoughts:
1. The 2 bows: It seems Viswakarmaji had designed two powerful bows, which he gave one each to Vishnu Bhagawan and Shiva Bhagawan. The one that Lord Rama drew and broke at the Sita Swayamvar was the bow of Lord Shiva. The one that Parasurama presents now to Lord Rama is the one belonging to Vishnu.
2. Parsurama was a partial avatar (others include Vedavyaasa, Kapila muni of Samkhya fame...) (I do think there is no surprise value in 2 avataras coexisting or overlapping because finally everyone and everything is Bhagawan as intelligent cause...The peculiarity of a amsha avatar (as Vanajamma said) is that ParaBrahman retains a part of the powers and sends a part to earth to manifest. (Sorry, that may sound very immature, but since it is understandable thus, I stated it thus. Then again, we know that everytime Bhagawan takes an avatra, it is for protecting Dharma. (Paritraanaaya sadhunaam
[The Srimad Bhagwatham explains the numerous avataras, amsha avatras, the avadhutas etc in painstaking details.]
3. Parasurama's story is long and deep. In brief his birth as avatar of Vishnu or let's say, Vishnu Bhagawan took the avatar as Parashurama in order to destroy the adharmic Kshatriya princes of the Sat Yug. There was a king named Karteeverya Arjuna whose sons killed Jamadagni, father of Parashurama...rich and deep story.. look up Srimad Bhagawatam...and to avenge his father's death it is said Parashurma cleaned up all the kshatriya kings of that time, save Janak and Dasaratha...
4. The important thing here would be the transfer of His powers. Parsuram as partial avatar had some of the powers, which he loses to or transfers off to Rama in that encounter. Let’s say, the powers devolved back to Vishnu in Ramaavatar, so that Parasurama as the avatar ceases to be the Power that is Vishnu. Maybe we can understand that as: The Parashurama avatar had served its purpose with the cleansing of the adharmik kshatriyas...so the powers were 'surrendered' almost...but read on..
5. I have also read that not all his powers dissolve back to Rama. Some remain which he does display during the Krishnavatar where he is seen as Karna’s teacher (a beautiful guru-shishya episode that is again unforgiving and harsh, but then that is again for us to learn something from...).
6. That Parasurama is seen in Ramavatar and again in Krishnavatar which are so far apart may be explained by the fact that Parasuram is a chiranjeev…
7. One more story goes (and I prefer this one
that when Rama broke the Shiva bow at Sita Swwayamvar, its thunder was heard by the sages all over the cosmos... and so did Parasurama too hear... he went to investigate and came across Sita-ma, and on meeting her he is said to have blessed her 'Deergha sumangali bhava!' so that when he encounters Rama, he keeps his vow and just surrenders.
But whatever the story, I think the moment in Time had come for the power of Vishnu in Parashurama to ascend into Vishnu as Lord Rama...
What remains difficu;t to comprehend are the feelings of despondency that Parashurama feels when he retires into Mahendranagar or Mahendrapuri (dont know)...but I guess this is a subtlety that can be taken as a story for our worldly ears...
I have emptied all my understanding as is, pls forgive me for any errors and haste.