Whenever the player walks/teleports away from a tamed mob (wolf, ocelot, parrot) at a certain distance where it will teleport, the mob will glitch out. It won't move from the place; if you have more than 20 fps, it will flicker left to right. But the player is able to interact with the mob - make it sit/stand, and pick up parrots on shoulders.
In the last case you can "duplicate" a parrot: you'll have it on your shoulder and a "ghost", not interactable version of it. Jumping will make a real parrot appear, but the ghost one won't disappear.