You could use a paper bag like terry suggested, colored construction paper works too-- you can "paint" on designs or messages with food dye [emoji41] and use sisal twine to tie the wrapping shut.
I've wondered about wrapping paper too. Out of curiosity, is it the way wrapping paper is processed that makes it bad for parrots? Or is it because of the type of paper and the color? Just wondered...
Presently have six Greenwing Macaw (17 yo), Red Fronted Macaw (12 yo), Red Lored Amazon (17 y.o.), Lilac Crowned Amazon (about 43 y.o.) and a Congo African Grey (11 y.o.)
Panama Amazon (1 Y.O.)
My mom's birds also shred Christmas wrapping paper. Not the metallic-foil like stuff, just the regular type. They like to unwrap YOUR gifts early for you
The dollar tree has some cute craft paper wrapping paper that has designs on it, but not overwhelming amounts of ink. That's probably a good option. But don't forget, you could always get a dud like Kiwi. He doesn't shred wrapping paper like other birds. Instead, he has you unwrap his gifts for him whilst eagerly nibbling your fingers as if to say "unwrap faster human!" I don't tend to wrap his gifts anymore, he doesn't appreciate it.