According to http://www.whateats.com/what-eats-a-parrot

The main predators, or natural enemies, of parrots are birds of prey, such as hawks, owls and eagles, and large snakes such as pythons and boa constrictors. Snakes grab parrots when they are resting on tree branches, while birds of prey attack them when they are flying as well as when they are at rest.

The black eagle of Asia often attacks and eats parrots and other birds when they are in their own nests!

Monkeys also eat parrot eggs and baby parrots whenever they can find them.