I have compiled a small list of open and recently-closed captive animal places (mostly petting zoos and farm parks) that aren't on the list in the OP; both for want of something better to procrastinate with doing, and on the off-chance that anyone here will be interested.
LEGEND:
◯ = Appears to still be operating
[✕] = Operating, but not usually open to visitors
✕ = Closed
✕ Action World Paihia (Bay of Islands): A family recreational facility that apparently had domestic ducks and a small number of domestic mammals. Closed in 2020.
◯ Crystal Mountain (Auckland): Includes a farm park; online photos show alpacas, emus, Indian peafowl, etc. I must note that the farm park is but one part of a "fascinating" combination of children's rides with copyright infringement, a mineral museum and shop, and a big collection of kitsch statues of extinct and extant animals. Would it be cruel of me to say that this could well be what Butterfly Creek will look like ten years from now? https://www.crystalmountain.co.nz/Theme+Park/Animal+Park++Rides.html
◯ Mini Golf Rotorua: Has domestic rabbits, and possibly a few common birds and fish. Mini Golf Rotorua
◯ Flat Hills Café & Tourist Park (Rangitikei): Has a petting zoo.
✕ The Alpaca Place (Masterton): Alpaca farm that was open by appointment; closed to visitors in mid-to-late 2020 or shortly afterwards.
✕ Gallin Farm Alpacas & Farmstay (Masterton): Alpaca farm that was open by appointment; closed to visitors in 2023.
[✕] The Black Sheep Animal Sanctuary (Ōtaki): Rescue facility for domestic animals. Occasionally has open days. About us
✕ Lotsa Follies (Tawa, Wellington): Petting zoo that seems to have closed in 2018 or shortly afterwards. Apparently may re-open, but whether this will eventuate remains to be seen.
◯ Trickle Creek (Horokiwi, Wellington): Very small-scale petting zoo that is open by appointment, but mainly intended for very young children. https://www.tricklecreek.co.nz/
◯ Te Paranui (Koromiko, near Picton): Perhaps technically a farm park, but seems to be more like a rescue facility for domestic animals that happens to be open to visitors on Saturdays. Visitors are not allowed to bring food that contains animal products of any kind. Te Paranui – Animal Sanctuary in Marlborough
◯ New Zealand Raptor Trust's "The Raptor Experience" (Timaru): A rescue facility for terrestrial birds of prey that is open by appointment. Appears to have very few permanent captive birds; possibly just two swamp harrier and two little owl. Intends to hold NZ falcon in the future but this may be unlikely.
Website: https://nzraptor.org/
Further reading: https://i.stuff.co.nz/timaru-herald/news/131892554/nz-raptor-trust-to-spread-its-speciessaving-wings
◯ The Animal Ark Dunedin (Mosgiel): Petting zoo.
✕ Mataura Petting Zoo (Gore): Closed in late 2019.
LEGEND:
◯ = Appears to still be operating
[✕] = Operating, but not usually open to visitors
✕ = Closed
✕ Action World Paihia (Bay of Islands): A family recreational facility that apparently had domestic ducks and a small number of domestic mammals. Closed in 2020.
◯ Crystal Mountain (Auckland): Includes a farm park; online photos show alpacas, emus, Indian peafowl, etc. I must note that the farm park is but one part of a "fascinating" combination of children's rides with copyright infringement, a mineral museum and shop, and a big collection of kitsch statues of extinct and extant animals. Would it be cruel of me to say that this could well be what Butterfly Creek will look like ten years from now? https://www.crystalmountain.co.nz/Theme+Park/Animal+Park++Rides.html
◯ Mini Golf Rotorua: Has domestic rabbits, and possibly a few common birds and fish. Mini Golf Rotorua
◯ Flat Hills Café & Tourist Park (Rangitikei): Has a petting zoo.
✕ The Alpaca Place (Masterton): Alpaca farm that was open by appointment; closed to visitors in mid-to-late 2020 or shortly afterwards.
✕ Gallin Farm Alpacas & Farmstay (Masterton): Alpaca farm that was open by appointment; closed to visitors in 2023.
[✕] The Black Sheep Animal Sanctuary (Ōtaki): Rescue facility for domestic animals. Occasionally has open days. About us
✕ Lotsa Follies (Tawa, Wellington): Petting zoo that seems to have closed in 2018 or shortly afterwards. Apparently may re-open, but whether this will eventuate remains to be seen.
◯ Trickle Creek (Horokiwi, Wellington): Very small-scale petting zoo that is open by appointment, but mainly intended for very young children. https://www.tricklecreek.co.nz/
◯ Te Paranui (Koromiko, near Picton): Perhaps technically a farm park, but seems to be more like a rescue facility for domestic animals that happens to be open to visitors on Saturdays. Visitors are not allowed to bring food that contains animal products of any kind. Te Paranui – Animal Sanctuary in Marlborough
◯ New Zealand Raptor Trust's "The Raptor Experience" (Timaru): A rescue facility for terrestrial birds of prey that is open by appointment. Appears to have very few permanent captive birds; possibly just two swamp harrier and two little owl. Intends to hold NZ falcon in the future but this may be unlikely.
Website: https://nzraptor.org/
Further reading: https://i.stuff.co.nz/timaru-herald/news/131892554/nz-raptor-trust-to-spread-its-speciessaving-wings
◯ The Animal Ark Dunedin (Mosgiel): Petting zoo.
✕ Mataura Petting Zoo (Gore): Closed in late 2019.