We don't use them over the whole floor, but washable splash mats usually have a grippy side and a wipe-able side, are made of plastic & fabric, and are non toxic. They're designed to go under high chairs and catch spilled food; I have one under each bird's cage & in areas where I think the birds might accidentally poop. They are not very comfortable to walk across though, so I wouldn't cover the entire floor with them. In the past I've also used hard plastic office mats designed to go under a desk, but I found that they were harder to keep clean (and the fact that the splash mats can be thrown into the wash when they get really gross is great).
I've had parrots in an apartment before (our Amazon, tiel, and a long time ago an african grey) with carpet and I've found honestly if you keep an eye on them and catch the poop while it's fresh, a bit of hot water and a towel will get it right out of most carpets with very little fuss. For worse spills or if you miss one and it dries, vinegar & baking soda should take care of it.
Another thing that worked well for us is using those removable wall hooks (like the Command hooks, etc) and putting a shower curtain up behind the cage to protect whatever wall it's against. Our birds are terrible about flinging chop and nothing sticks to paint like bell pepper skin!
