Where did the common pop-culture way of portraying pirates (yarr!) come from? Did pirate captains commonly have birds on their shoulders?
Piratespeak. "Arrrr" showed up late, probably in movies of the 1930s. Actor Robert Newton played Silver in the 1950 version of Treasure Island, one of the better portrayals of old-school piracy, and reprised the role in sequels and on TV; his accent featured a strong rolling R, which likely helped fix "arrrr" in the piratical canon. Much pirate lingo, like "avast," was simply nautical speech of the time; "shiver my timbers" predates Stevenson, but he ran with it.
To specifically prove "avast" mentioned above, http://en.wiktionary.org/wiki/avast says:
Etymology: From the Dutch hou' vast (“hold fast”), from houd (“hold”) + vast (“fast”)
As far as parrots - the above linked straightdope article didn't really have enough level of details, but this blog does: http://pirates2.sakraft.com/labels/History.html (Of Pirates and Parrots; Thursday, June 19, 2008) as well as "Parrots in History" page which quotes from that blog and adds its own info:
Pirates and Parrots?
By the time of the Golden Age of Piracy (1680 – 1730) there was a well established trade in parrots. Usually animals aboard a ship were used as provisions rather than as pets but fortunately parrots weren’t a favorite meal.
It is thought that the pirate & parrot cliché originated from Robert Louis Stevenson’s “Treasure Island”.
However, the historical journal of Pirate Captain William Dampier describes that in certain cases pirates actually did take parrots on board, most likely because they wanted to sell them at a profit to the high society of Europe. In Dampier's journal it is mentioned that parrots were stored along with other animals and provisions on the ship, while anchoring at a Caribbean island:
"The tame Parrots we found here were the largest and fairest Birds of their kind that I ever saw in the West Indies. Their colour was yellow and red, very coursly mixt; and they would prate very prettily; and there was scarce a Man but what sent aboard one or two of them. So that with Provision, Chests, Hencoops and Parrot-Cages, our Ships were full of Lumber, with which we intended to sail."