Elephant rides disappeared in Australia in the early 1960's. At my local Melbourne Zoo they stopped in 1962, too early for me to have a personal memory of them. A few years back the new Director of one of our major zoos suggested in a flurry of enthusiasm for his new job that he would reintroduce elephant rides. Somebody must have had a quiet word because that was the last heard of it. This Director does not have an animal or zoo background.
Melbourne Zoo had pony rides I think until the early 1980's, attached to a small amusement park that operated within the zoo. The park was operated by a lessee, and the ponies disappeared with the rest of the park when the zoo decided not to renew the lease.
Dromedary camel rides are not uncommon in Australia, hardly surprising given how common feral camels are. However I am aware of only one Australian zoo that has them. It is a large privately owned Australian wildlife park which has a "farmyard" attached, and the camel rides are part of the farmyard. However again I understand the farmyard is operated by a third party.
There are a number of petting farms that also operate pony rides etc. While there is debate as to whether these qualify as zoos, many have small collections of Australian and/or exotic wildlife.