With birds you actually need to be more careful about squeezing their chest than their neck. You need to be careful with both obviously, but what I was told by a vet tech is that parrots have a bone in their throat that humans don't have, which helps prevent them from being choked by pressure on the neck. But they don't have a diaphragm like humans do, and they MUST be able to move their chest up and down in order to breathe.
Edited to add: I did some quick research and found a picture that shows this bone pretty well, sticking out like a ring at the base of the throat. It's actually the wishbone, aka furcula. Obviously it doesn't protect the entire neck, but I can see how it would help prevent the lower part of the neck from being compressed. I can see why they might need to move that massive keelbone in order to breathe too.