List of Australian Parrots:

Australian Ringneck (Barnardius zonarius)
Australian King Parrot(Alisterus scapularis)
Blue Bonnet (Northiella haematogaster)
Bourke's Parrot (Neopsephotus bourkii)
Budgerigar (Budgie or Parakeet) (Melopsittacus undulatus) Budgie FAQ
Eclectus Parrot (Eclectus roratus)
Fig Parrot - Double-eyed Fig-Parrot (Cyclopsitta diophthalma)
Golden-shouldered Parrot (Psephotus chrysoptergius)
Hooded Parrot (Psephotus chrysopterygius)
Mulga Parrot (Psephotus haematonotus)
Night Parrot or Snub Nosed Parrot (Pezoporus occidentalis)
Port Lincoln Parrot (Barnardius zonarius)
Princess Parrot or Princess of Wales Parakeet (Polytelis alexandrae)
Red-capped Parrot (Purpureicephalus spurius)
Red-rumped Parrot (Psephotus haematonotus)
Red-winged Parrot (Aprosmictus erythropterus)
Regent Parrot or Rock Pebbler (Polytelis anthopeplus)
Superb Parrot or Barraband, Green Leek (Polytelis swainsonii)
Swift Parrot (Lathamus discolor)

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Cockatoos:
Cockatoo FAQ
(Probosciger, Calyptorhynchus, Callocephalon, Eolophus, Cacatua, Nymphicus)

Baudin's Black Cockatoo (Calyptorhynchus baudinii)
Carnaby's Black Cockatoo (Calyptorhynchus latirostris)
Cockatiel (Nymphicus hollandicus)
Galah (Eolophus roseicapilla)
Gang Gang Cockatoo (Callocephalon fimbriatum)
Glossy Black Cockatoo (Calyptorhynchus lathami)
Little Corella (Cacatua sanguinea)
Long-billed Corella (Cacatua tenuirostris)
Major Mitchell's Cockatoo (Cocatua leadbeateri)
Palm Cockatoo (Probosciger aterrimus)
Red-tailed Black Cockatoo (Calyptorhynchus banksii)
Sulphur-crested Cockatoo (Cacatua galerita)
Western Corella (Cacatua pastinator)
Yellow-tailed Black Cockatoo (Calyptorhynchus funereus)

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Grass Parakeets:
(Neophema):

Blue-winged Parrot (Neophema chrysostoma)
Elegant Parrot (Neophema elegans)
Orange-bellied Parrot (Neophema chrysogaster)
Rock Parrot (Neophema petrophila)
Scarlet-chested Parrot (Neophema splendida)
Turquoisine Parrot (Neophema pulchella)

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Lorikeets:
(Trichoglossus, Glossopsitta, Psitteuteles)

Musk Lorikeet (Glossopsitta concinna)
Little Lorikeet (Glossopsitta pusilla)
Purple-crowned Lorikeet (Glossopsitta porphyrocephala)
Rainbow Lorikeet (Trichoglossus haematodus)
Red-collared Lorikeet (Trichoglossus rubritorquis)
Scaly-breasted Lorikeet (Trichoglossus chlorolepidotus)
Varied Lorikeet (Psitteuteles versicolor)

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Rosellas:
(Platycercus)

Adelaide Rosella (Platycercus adelaidae)
Blue-cheeked Rosella (Platycercus adscitus)
Crimson Rosella (Platycercus elegans)
Eastern Rosella (Platycercus eximius)
Green Rosella (Platycercus caledonicus)
Northern Rosella (Platycercus venustus)
Pale-headed Rosella (Platycercus adscitus)
Western Rosella (Platycercus icterotis)
Yellow Rosella (Platycercus flaveolus)

I didn't think eclectus' were australian parrots, they're from the Solomon islands aren't they?

Well.. The Solomon island subspecies are from there, but the other subspecies scatter through Australia as well as a couple other islands. Just wiki it

Well you learn something news every day do they look the same? Are there any in the wild?

Gotta catch 'em all!