There use to be (6 years ago), so I'm sure there still is, a private facility in Dural (western Sydney) which hold a number of primates. When I visited there in 2002, the woman who owned it had Spider Monkeys, Pig-Tailed Macaques, Rhesus Macaques and maybe one or two other species.
Plus I'm not sure if the old, now closed, Mugga Lane Zoo in Canberra still has any old stock.