1) Its not from a pirate-themed anything, but there was an African Grey Parrot in a movie called It Waits. Not a very good movie at all- the Parrot has all the best lines.

For example, unlike the humans around it, it can actually tell when the movie's monster is around (and obscured from sight)...and says things like "Uh Oh!" in its cutesy parrot voice. The humans never really catch on- at one point, the heroine actually tries to set him free for his safety...nevermind that he's an ideal warning device! It has other lines as well, but that stood out to me.

(Personally, I'm dying to use it for a PC of my own some day...but it would be a nice familiar or animal companion for a pirate (N)PC.)

Other whacky takes on the old parrot-on-the-shoulder pirate schtick would include birds like Ostriches, Axebeaks, Penguins, or even some of the smaller dinosaurs (esp. the raptors and birdlike ones).

2) I once started a campaign with an interdimensional raider themed twist on the old Against the Slavers modules- the party's transport was attacked by transdimensional pirates and eventually taken captive.

While it was fun, I'm not suggesting that as a starting point for your campaign.

What I'm suggesting is that you take a close look at the way pirates capture ships and do at least one attempted ship capture- its a great big melee with 3 dimensional consequences that also involves strictures. Most seamen of all kinds didn't even know how to swim, so once they hit the water...

3) Use superstitions. There are all kinds of mysterious beasties below the waves and beyond the horizon, and everyone knows about ghost ships and the restless spirits of men lost at sea.

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