That's an interesting question! If you can find a good diagram of a parrot skull that demonstrates the craniofacial hinge, you'll see it's almost vital to the good operation of the bird's gape. I suspect all parrots probably do have craniofacial hinges, but I'm not 100% sure. Wouldn't you know it, but not one of my anatomy books has the answer, either! They're all general vertebrate anatomy and not bird-specific.
My best suggestion would be to contact either your avian vet or the Zoology Department at a local University. When you find the answer, please post it so I can know too!