I am curious why would you ask that? Parrots look quite different to most birds, don't you think the OP would know the difference?
I think it is a Monty Python reference..
No, honestly. Common sense please. This is to avoid pollution of water courses, etc- and intended for larger animals. It is true that you are no longer allowed to bury large animals in CH, for that reason. A small bird you can bury yourself in a hedge or wood, no problem! and no pollution, honestly.
Having said that, when our elderly cat and dog depart this earth, they will stay in a corner of our field. Call the cops if you wish.