...Based on Valmiki Ramayan.
The bow which was in Janaka's custody and which was strung / broken by Rama was originally's Shiva's and was handed over to Janaka's predecessor- King Devavrata.
Shiva and Vishnu once had a battle and in the battle Vishnu uttered the sound "Hmmm" due to which the string on SHiva's bow got loosened and he could not tie it back. Vishnu was declared the winner. A disappointed Shiva handed the bow to King Devavrata- the then ruler of Mitihila from whom it got inherited by each successive generation, until King Janaka.
This was the bow which Rama strung and broke.
In the meanwhile Vishnu handed over his bow to Sage Richika and from him it passed on to Sage Jamadagni and from him to Parashurama. This was the bow with which Parashurama challenged Rama on their way back to Ayodhya.
Actually, Parashurama is not angry in this situation- though all the serials have depicted him thus.
It was just that Parashurama had a rather fearsome appearance. His gigantic body, his matted hair and the powerful axe in his arms- all added to his ferocious looks. And whenever he chose to appear, it was through a whirlwind.
Because of his reputation and his known hostility towards Kshyatrias, everyone trembled in his presence. Dasharatha was no exception.
Parashurama was merely intrigued to know who could have broken Shiva's bow- since he, Parashurama, had its counterpart, the Vishnu Bow. It could not have been an ordinary person since the bow had remained unstrung for many generations in Janak' s family.
No where (in Valmiki Ramayan) is it mentioned that Parashurama felt insulted that Shiva's bow had been broken. He was surprised and amazed. He just wanted to doubly ensure that it was indeed Rama who had broken the bow. So Parashurama offers him the counterpart bow- the Vishnu bow which was equal in all respects and asks him to place arrow and stretch the string up to his ear.
Rama, being Vishnu himself could do so easily. Parashurama realized that it was indeed Narayana in human form, since none but Narayana could so effortlessly wield both the bows. Parashurama has no hesitation in sacrificing his ascetic merits since he was convinced that his purpose on Earth was over. Another form of Narayana had come to take over from where he had left.
After Parashurama vanished, Rama held the bow aloft in his hand, and Varuna (from the heavens) collected it.
The bow with which Rama fights and kills Ravana is the one given to him by Sage Agastya (along with divine arrows and two inexhaustible quivers), though the name "Kodanda" is not mentioned anywhere in VR, at least to the best of my knowledge
Hope this helps.