How big was the parrot?
Maybe I have the wrong picture in my head, but bird cages I've seen (unless it's like an aviary) are too small to do much composting in.
Otherwise probably ok. My compost bin is a 4' cube of wire bars in a grid.
Works fine. The grid is not small enough to keep little creatures (e.g. mice, some birds, etc) out, but it is small enough to keep big creatures (e.g. groundhog and raccoon) out.