When an untamed parrot gets its name changed, all players see it.
But if the owner of a tamed parrot renames their parrot, the name does not show up for other players UNTIL the parrot mounts and then un-mounts from the owners shoulder.

However, if a player who does NOT own the parrot renames the parrot, the parrots name change will show up for both owner and non-owner.

This bug only applies to Parrots, not Ocelots/Wolves.

This bug was tested with a friend. But for convenience sake, I used my account running on the same computer.