The bug
When you put an invisibility potion on a parrot and then put it on your shoulder, it turns visible.
Linked issues
is duplicated by 3
relates to 4
MC-115038
Upside down ("Dinnerbone" / "Grumm") parrots not upside down as ShoulderEntity
Resolved
MC-114896
Parrots don't make any sounds when they are on your shoulder
Resolved
MC-115048
Parrots sitting on shoulders don't drown in water or burn in lava / Parrots shouldn't go diving
Resolved
MC-123607
Parrots can still be sitting on shoulders, when they are burning, poisoning and withering
Open
Seems at first glance intentional to me, from the way the parrots are currently implemented as ShoulderEntity, however fixing this bugpost here should be made in logical consequence to the decision in how to solve e.g. MC-115048.