I've seen a lot of rabbits running around at the Houston Zoo, though I don't know if the zoo put them there or if they're wild animals taking residence.
Hamburg still has the free-ranging Reeve's muntjacs, maras and capybaras. Eastern chipmunks at Wuppertal Zoo. Griffon vultures and ring-tailed lemurs at Zoo Salzburg. Juvenile colobus monkeys at Zoo Krefeld. Introduced ring-necked parakeets in various European zoos. Yellow-headed amazons at Stuttgart (including Wilhelma) Squirrel monkeys at Tierpark Hellabrunn (Munich) Golden-headed lion tamarins at Zürich Zoo Cotton-top tamarins at Jihlava Zoo
Zoos with a focus on local species like Innsbruck Zoo, wildparks, reptile zoos...I've seen a few without any peafowl.
On my last visit to Edinburgh, I spotted a robin at the bench outside the gift shop. And there were a blackbird in the rhea enclosure. When I left, the robin was still there.
Many German and Danish zoos have free-ranging domestic cats. Karlsruhe Zoo has free living Great white pelicans and Zamosc Zoo in Poland has even free-ranging Spotted sousliks. Wuppertal Zoo has - at least when I visited - quite confiding Eastern chipmunks. Peafowls, ducks etc are - I guess so - free-ranging in every bigger zoo. In some zoos I saw how some goats or sheep escaped from the children's zoo because some inobservant guys let the doors open.
Phoenix Zoo has a variety of lizards and waterfowl and snakes. They have free-range Helmeted Guinea Fowl and have several signs telling visitors that the Guinea Fowl are not escapees. And while the Pea Fowl do not belong to the zoo, they have been banded. Not real sure why, other than to maybe alert people who see them outside zoo grounds that they mostly live there. There's a water/utility company near the zoo and when my mom worked there she would often see the pea fowl hanging out in the company's fountain
I know of a few Danish zoos that keep them in aviaries instead. I don't know whether it's for the benefit of the visitors (not that free-ranging peafowl are hard to see, but they're even easier to see if they're confined to a small space) or they just didn't consider to let them out. One of them, Munkholm Zoo, has free-ranging housecats so maybe the peafowl are confined to protect the chicks. Aalborg Zoo doesn't actually have peafowl anymore, since the last one, a free-ranging male, entered the old tiger exhibit a few years ago and got eaten.
There are literally hundreds of species which could be suitable. I'm not that inclined to produce a list...