Many of the animals were allowed to freely roam the park. Deer, goats, ducks, geese, and other birds had the run of the land for the most part. The way the hoofed animal enclosures were, it looked like they could selectively open the gates to allow certain animals to roam while others were stuck in place.
The more dangerous animals - bear, lynx, wolves - and the tiny critters - rabbits, ferrets and the like - were stuck in enclosures.
I don't think they let all the animals out at the same time, so it's possible that on different days they might let the pigs out or maybe the buffalo. I only saw deer-type critters and birds running freely.