I have a friend who did something similar - she uses old t-shirts cut into strips and then puts them around a circular thing (such as a hula hoop) and voila, cheap swing (just going to add, yes hula hoop but she has two large macaws...)
Why old rather than new? Well it's a new use for a shirt you are no longer going to wear
Personally, my old shirts become "bird shirts", aka shirts I wear around the birds since I like to avoid any, hmm, "accidents" they might have while on me on any clothing I might want to wear out