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Around the World in 80 Dreams?
It’s a British animated film called The Talking Parcel. Made by Cosgrove-Hall, the same studio responsible for Danger Mouse, et al. It frequently turns up on DVD at dollar stores.
Also, someone posted the entire film to Youtube: https://www.youtube.com/watch?v=oQoQOpug3Wg
That's the one! Thank you! Didn't know it was also on "Long Ago and Far Away," as well; that's probably where I saw it for the first time.
I KNOW I saw this cartoon in school once, around the early 90s/late 80s. It was about a girl who found a parrot (who I think was simply called "Parrot") in a floating box at the beach, and went on adventures with him. Along the way they met a sea monster with a hearing horn that was friends with the parrot, and some frog-creature who was good at imitation. (Maybe through the use of paper thin disguises, or actual magic; I can't remember much on that.)
I know that there was also a werewolf they had to trick at some point, and the big bads were some sort of giant-bird-THINGS that had taken something Parrot needed to get back. And somewhere in the final battle there were unicorns and a dragon.
Pretty sure this was also a one-shot film, and not a series. I'm wracking my brain for more, but that's all I can come up with. Sound familiar to anyone?