During the scene in which Peter's parrot gets injured, the parrot says "shoot me." This is a reference to a scene in the film Full Metal Jacket in which a sniper is killed.
After Peter's parrot dies, he says that he screwed up worse than Disney did in casting Michael J. Fox in a Zorro remake, a reference to the box-office failure of The Legend of Zorro and Fox's Parkinson's Disease as indicated by the messed-up Z insignia. It should also be noted that "zorro" is Spanish for "fox."
When Peter and his pirate friends break into Goldman's Pharmacy, Mort yells "Oh, no! Kristallnacht!" Kristallnacht was when the Nazi soldiers destroyed many Jewish-owned stores.
The 80's novelty answering machine rap message is a reference to a real cassette tape that contained funny answering machine greetings called Crazy Calls.