Tamed parrots land close to the player and stop flying. When the player moves, they only move their heads in his wake, and if the player moves a little further, they take off, sit close to him again and stop moving. Because of this they rarely sit on the player's shoulder. Even if there is a hostile mob nearby that they can imitate, they only repeat its sounds while continuing to stare at the player, i.e. do not help the player find the source of the sound. if the tamed parrot is sitting and the player is far away. then the parrot does not move at all, and its head looks forward.
how it must work
Tamed parrots imitate hostile mobs sounds and also look at their location. For example, a parrot imitates a zombie that is underground, not far from the parrot, then the head of the parrot should be turned towards this zombie