When the player falls from a great height in survival mode with a parrot mounted on their shoulder they do not receive any fall damage. I first noticed this when I was playing in survival and fell in a ravine.

Steps to reproduce:
-Tame a parrot.
-Let it mount on your shoulder.
-Jump off from a high place

[Note that the parrot does not dismount]
Doesn't matter if you started the world from creative or survival. The results will be the same.