I recently created a similar issue ('sitting parrots fly if hostile mobs are nearby') which was resolved as 'cannot reproduce'. I have done some more testing and the issue seems to be that tamed sitting parrots will fly if a hostile mob nearby does damage to you. This is inconvenient because the parrots swarm you and it's very hard not to kill them while defending yourself from the hostile mob.
Linked issues
duplicates 1
Comments 3
It's also not just if they are on your shoulder. (Not sure if I made that clear.) If they have been made to sit on e.g. the ground nearby they will also fly if you happen to take damage.
It looks like this is caused by MC-696.
This is described as intended behavior, see this comment. However, I agree that this might not be a great solution in combat.