In the golden age of piracy yes, pirates had pet monkeys, parrots, and other exotic animals because pirates were a part of the trade in exotic animals. Sometimes they were just transporting the animals, and sometimes they grew attached and kept them. Laxer standards of behavior allowed for this to be more common on pirate vessels than on merchant men or navy vessels, although it was not unheard of on these kinds of ships as well, particularly amongst higher ranking seamen.

Other parts of the pirate mythos were fabricated by books, plays, and movies. For example, we associate peg legs with pirates because of Treasure Island's Long John Silver and pirates having hooks for hands came from Peter Pan's Captain Hook. Although pirates occasionally lost body parts in battle (all surviving pirate articles included compensation for list limbs and eyes), they were not able to remain pirates afterwards since pirates were, first and foremost, sailors. As scary as the idea of a hook hand is, it's hard to climb the rigging and tie knots without two hands. Peg legs, similarly, have their detriments when attempting to climb rigging and walk across spars. You may notice that Captain Hook was a captain, and Long John Silver was the ship's cook, both positions that are infrequently called upon to do the normal work of sailing (although pirate captains were often considered the first among peers and might participate in the duties of sailing the ship, depending on their personal preference and relationship with the crew).

Source: A History of Piracy by Philip Gosse