Parrots with the invisibility effect have their model rendered anyways when they are on your shoulder.

Side note I tried to get the data of the parrot whilst it was on my shoulder and the command didn't find an entity, so I think what happens is that when it goes on your shoulder the entity is temporarily removed from the game and that's why it happens, supporting this theory, other potion effects like poison don't take effect while it's on your shoulder as well.