The Parrot in Eric
A supposedly talking bird, 50% or more of whose vocabulary revolved around one metasyntactic variable, whom Eric inherited from a late relative along with the latter's books on demonology and necromancy.
In appearance resembling a mis-used toilet brush with half the bristles missing, the Parrot was given to Ponce da Quirm just before Eric and Rincewind ran away, very quickly, in the Tezuman Empire. The parrot accompanied da Quirm to Hell, where it adds an extra dimension of exquisite torment to his afterlife.
Parrots in general (though not this specific parrot) were used in Moving Pictures to add sound to movies. This failed miserably, in part because the handlemen (cameramen) taught them to say things like "show us your knickers"... and the Discworld wasn't *quite* ready for films like that (yet).
The Parrot in Nation
A parrot plays a subtle linking role in Nation. Originally the ship's parrot aboard the Sweet Judy, after that ship founders on the tsunami which wrecks the original Nation, it escapes to the forest and enjoys itself by persecuting the Grandfather Birds. Taught to use most unbecoming language by Cox, Polegrave and Foxlip, as part of their campaign to undermine the authority of Captain Nathan Roberts, it announces its presence with gleeful cries of Show us your drawers!, and other similar remarks which would have sent Daphne's grandmother into a fit of the blue vapours. The Parrot can perhaps be viewed as symbolic of the forces of nature, sweeping aside established order. The British ship is wrecked with almost all aboard her, but the Parrot lives; the thousands of years of ritual and taboo that ruled the Nation, represented by the Grandfather Birds, are similarly batted aside with contempt by something new and unstoppable.
Fittingly (see Annotation below), a determined effort is made by the mutineers to kill the parrot, which at first looks as if it has succeeded. (Cox, in the event, only wings it). Well, you can't have a Parrot in a Pratchett story without it having at least one brush with death, can you?
Sending a dead parrot to its afterlife was probably irresistible to one who is Monty Python-literate.
He has rung down the bleedin' curtain and joined the choir invisible! He has snuffed it! He has gone to meet his Maker! This parrot is no more! This.... is an EX-PARROT!