While the image is definitely a Hollywood thing and it’s unlikely that it was true, the origin is from two books. The first, released at the end of the Golden Age of Piracy itself: the semi-true but mostly fiction “A General History of the Robberies and Murders of the Most Notorious Pyrates,” a 1724 book whose author wrote under the alias Captain Charles Johnson. This book is filled with historical inaccuracies which do not match up with shipping logs, historical events or people’s names, so anything in it is dubious.

The second was Treasure Island, 1883, by Robert Louis Stevenson. It is a complete work of fiction but most specifically mentions a parrot on a captain’s shoulder.

While it is possible that some of the more flamboyant pirates such as Stede Bonnett or Jack Rackham kept a parrot as a pet, it’s somewhat unrealistic that it would be on their shoulders as they are in the movies