I do not know of any historical accounts mentioning the pets of known pirates. However, there are many fictional accounts of pirates with pets and we can assume that these probably have some basis in fact. Still, from the historical evidence we can infer that Pirates would have had pets, just as other people did.

Historically, we know the following about parrots and human co-habitation. Parrots were first brought to ancient Greece. At the court of Artaxerxes II, the physician Ctesias gave a description of the Plum-headed Parakeet and noticed it's ability to speak.

Parrots from the genus Psittacula were also well known in ancient Rome.

With the longer ocean voyages around the coast of Africa and later the discovery of the Americas,the trade in live parrots became a regular business.
By the time of the Golden Age of Piracy, there was a well established business in exotic animals. It was not uncommon for royalty in Europe to own live animal menageries. Henry I of England established his Royal Menagerie in around 1125. Philip VI of France founded the Paris Menagerie in 1333. Louis XI (1461-1483) founded the Touraine Menagerie and Louis XIV (1643-1714) the Versaille Menagerie. The actions of the Heads of Europe, were most certainly be mirrored by the lesser members of the ruling class.

The more exotic the animal, the more in demand and the more valuable the animal would be. A talking bird would fetch a few doubloons in the Royal Courts of Europe, as would New World Monkeys. We also know that if something was of value, then pirates would plunder it. We also know that some pirates were known for their garish costumes. Whatever was good enough for the royal courts was also good enough for the well-to-do Pirate. To own an exotic animal would have been a status symbol.

A parrot's social behavior has always made it a good pet. They seem to instinctively accept humans as one of their flock which makes touching and holding the bird possible. They may learn to imitate human voices. Their ability to learn tricks and behavior which is not typical for the wild bird have also made them popular with humans.

Other animals such as common cats and dogs may have been kept as they have been co-habitating with humans for thousands of years. The animals would have served the same practical purposes they do today. Early wood-block illustrations provide evidence that Bucanneers indeed kept hunting dogs. There is also evidence that domestic animals were taken on voyages. Some scientists believe that the extinction of the Dodo Bird is directly related to the introduction of pigs, cats, and dogs to the Mauritius Island, the birds only habitat. We can assume that what was good enough for merchant ships was also good enough for a pirate ship.
Pirates were also a superstitious lot and the possession of black cats was considered good luck among British merchant marines during this time period (and still today in some circles). It was a common practice for the loved ones of a merchant marine to keep a black cat in the house while he was at sea. The belief was that so long as the cat was well fed and kept safe from harm, nothing would happen to the mariner.

Sailors also kept cats on board ships to bring them luck.Other superstitions involving sailors and cats claimed If the ship's cat approached a sailor, it meant good luck. If the cat approached halfway and went away, bad luck would surely follow. If a cat was thrown overboard, a storm would rise and very bad luck would follow.

There is no reason to doubt that an English pirate or his lover would not hold the same belief as that of other merhants of the sea.