Did you have the parrot before you were with the fiancé?
I can somewhat understand if the relationship came first. There can be an element of "I didn't sign up for this", and people are allowed to have dealbreakers.
...
Fiancé can go sit on a tack, and I would consider this the reddest of flags.
If you date someone, you recognize they're a package deal with the things they love, be it kids or pets or their career or hobbies or whatever.
Telling them "choose between me and the thing you love, that I knew about from day one?"
Holding the relationship hostage to get what they want?
Good, kind people don't do that. It's controlling and manipulative and, frankly, a precurser of what you can expect from the relationship in the future. I would have a long hard think about whether you want to marry this thief of joy.
Good, kind partners say "I want your bird to like me. How can I make that happen? What if I give him lots of treats and positive attention? Ask your bird forum... maybe they'll have some ideas.
If you had the bird first, then fiancé is showing you who they truly are, maybe for the first time. Believe them.