When you put an invisibility potion on a parrot and then put it on your shoulder, it turns visible.
Parrots can still be sitting on shoulders, when they are burning, poisoning and withering
Parrots don't make any sounds when they are on your shoulder
Upside down ("Dinnerbone" / "Grumm") parrots not upside down as ShoulderEntity
Parrots sitting on shoulders don't drown in water or burn in lava / Parrots shouldn't go diving