My pet parrot definitly laughs. It sounds like a staccato clucking sound. He usually does it when he's lying on my belly playing with me. He lays on his back, both feet grabbing my finger (or fingers) and play-biting my finger with his beak. I've gotten to where I can do a pretty good imitation which I do when he's clearly playing. (he does sometimes get too agressive in his play, and I usually stop clucking when he pushes too far, but I doubt that registers with him.)

EDIT: Just for completeness, he's a Mitered Conure. Conure's are probably the most gregarious parrots I've encountered.