then there can be no proper answer. If by "animal" you actually mean all animal species (as opposed to just mammals, or just mammals and birds) there would be probably between 500 and 600 species in NZ zoos, parks and aquariums. Individual numbers nobody could say: you would have to assume an average number per species which would not be very accurate. E.g. there is only one Asian elephant in NZ, but probably hundreds of guineapigs or chickens.
If you only mean mammals I could probably muck up a rough total but not for all animals.