Parrots start flying when the player takes damage from a mob, even when sitting.

Tamed parrots will start flying when the player takes damage from a hostile mob (tested with zombie, skeleton & spider), even when it has been sat down.

To reprocude, summon a few parrots and tame them using seeds. Make sure they are sitting down and not following behind you.

Summon a hostile mob (e.g. Zombie) and let it attack you. Observe how all the parrots are starting to fly the moment you get hit by the mob. they will sit down again wherever they land.

gif of the bug: https://gfycat.com/ScrawnyMedicalKagu
(can't upload because filesize to big)