Pirates would sometimes capture parrots or monkeys due to the price they fetched back home. During that time period, tropical animals were much more novel and unique than they are today. It isn't generally believed that many pirates actually kept them as pets though. This image most likely began with the book 'Treasure Island'. In the book, the ever famous Long John Silver carried a parrot as a pet. If anything, they would have been more likely to become fresh meat on a more difficult voyage.
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Yes, but they also had pigs, dogs, and cats, just to name a few. They had them for practical reasons, for example monkeys could be trained to get things from small places, cats could catch rats and mice on board, and dogs were naturaly loyal and made great friends during long sea voyages.
There were parrots, but they weren't pets. They were "investments" and were used to sell when they needed extra cash.
The answer is, They don't. They use them for speaking if they can't and, sometimes, for company. Yes they get terribly on your nerves, they say they kill them if they get too frustrating.