Why do you cut off a parrot's wing?

You don't. Such an act would be incredibly cruel. The only circumstances under which a parrot's wing would be removed would be if a veterinarian determined it had a cancer or similar, inoperable condition on or within the wing.

Many people with pet parrots trim back the flight feathers on one wing. this prevents the parrot from flying away, and if done properly, is effective and painless.