I took a look at your puppet photos. Love the soccer ball and turtle. I think the turtle would be a good place for you to start with the bird. The shape of the top of the puppets head would be a good shape for a birds head. Take a look at Axtells Cackatoo http://www.axtell.com/cockatoo.html as well as Pavlovs
http://www.pavlovspuppets.com/cockatoo_green.htm. Notice that the beak is an exaggerated version of a real beak. If you look at Axtells the mouth plate is wider at the hinge and curves to a semi point in the front.
These photos are of a longer beak, but all you'd need to do is shorten the beak and round the pattern up more to create the look of the Cockatoo's beak.
I have another photos of a Flamingo/Parrot that might be helpful.

Again...shrink down the size of the beak and round the foam more to create the more curved beak.
Julie