The taxonomy of the parrots has recently undergone a number of drastic changes, and the field is perhaps still a bit in flux.
What is listed here is a maximalist's presentation, in the hope of raising the fullest possible awareness amongst birders.

Not visible in this page is the split off of the Cockatoos as a separate family from the Parrots.

Uvea Parakeet is not recognized as a full species in HBW 4 but it "may merit treatment" as such.
 

Uvea Parakeet     Eunymphicus uvaeensis
(Layard & Layard) 1882
n New Caledonia: Loyalty Islands, viz. Ouvea Island
split from: Eunymphicus cornutus
insert after: Eunymphicus cornutus
N.J. Collar & M.J. Crosby & A.J. Stattersfield,
Birds to Watch 2: the World List of Threatened Birds. BirdLife Conservation Series 4, Cambridge, UK, 1994
 

Not recognized as a full species in HBW 4, although it is "sometimes considered a separate species."

New Caledonian Parakeet     Cyanoramphus saisseti
Verreaux & Des Murs 1860
New Caledonia
split from: Cyanoramphus novaezelandiae
insert after: Cyanoramphus novaezelandiae
Tony Juniper & Mike Parr,
Parrots---A Guide to the Parrots of the World, Pica Press, 1998, p. 357
 

The nearly extinct Chatham Parakeet is genetically distinct from the mainland Yellow-fronted Parakeet, C. auriceps.  The biggest threat to its survival it genetic swamping through hybridization with the chathamensis race of the Red-fronted Parakeet.

Actions against this involve culling of the latter and their hybrids, as well as habitat improvements.
This form is not recognized as a full species by Juniper & Parr.

Chatham Parakeet     Cyanoramphus forbesi
Rothschild   1893
off e New Zealand: Chatham Islands, viz. Mangere Is. & Little Mangere Is.
split from: Cyanoramphus auriceps
insert after: Cyanoramphus auriceps
S.J. Triggs & C.H. Daugherty,
Conservation and genetics of New Zealand parakeets
Bird Conservation International 6, 1 (1996): 89-101
 

Malherbe's Parakeet has been considered a colour morph of Yellow-fronted Parakeet, but should probably best be considered a separate species as some mechanism of assortative mating seems to be in place.

Malherbe's Parakeet     Cyanoramphus malherbi
Souancé   1857
c New Zealand: n South Island (formerly much more widespread)
split from: Cyanoramphus auriceps
insert after: Cyanoramphus forbesi
S.J. Triggs & C.H. Daugherty,
Conservation and genetics of New Zealand parakeets
Bird Conservation International 6, 1 (1996): 89-101
 

About this taxon, Nigel Collar writes in HBW4: "sometimes considered worthy of species status, based on black bill and grey legs, although these colours may be variable."

Black-billed Hanging-Parrot     Loriculus bonapartei
Souancé   1856
sw Philippines: Sulu Archipelago
split from: Loriculus philippensis
insert after: Loriculus philippensis
Tony Juniper & Mike Parr,
Parrots---A Guide to the Parrots of the World, Pica Press, 1998, p. 396-397
 

Not recognized as a separate species in Juniper and Parr.

Sula Hanging-Parrot     Loriculus sclateri
Wallace   1863
ec Indonesia: Sula & Banggai Island, off ce Sulawesi
split from: Loriculus amabilis
insert after: Loriculus stigmatus
Nigel J. Collar,
in: Handbook of the Birds of the World. Vol. 4, J. del Hoyo, A. Elliott & J. Sargatal (eds.), Barcelona, 1997
 

Also recognized as a full species in HBW4.

Socorro Parakeet     Aratinga brevipes
(Lawrence) 1871
nw Mexico: Socorro Island
split from: Aratinga holochlora
insert after: Aratinga holochlora
Tony Juniper & Mike Parr,
Parrots---A Guide to the Parrots of the World, Pica Press, 1998, p. 439
 

Not recognized as a separate species in Juniper and Parr.

Red-throated Parakeet     Aratinga rubritorquis
(Sclater-PL)   1887
e Guatemala; El Salvador s to; Honduras; n Nicaragua. Highlands
split from: Aratinga holochlora
insert after: Aratinga brevipes
Nigel J. Collar,
in: Handbook of the Birds of the World. Vol. 4, J. del Hoyo, A. Elliott & J. Sargatal (eds.), Barcelona, 1997
 

The Jamaican Macaw was predominantly red, but differing from A. tricolor in having a yellow, not red, forehead. Unfortunately, however, the type specimen appears to be lost.

Peters (1937, p. 184) states that this species was based on the "yellow-headed macaw" in Philip Henry Gosse's The Birds of Jamaica (J. van Voorst, London 1847, p. 260), who took his description from Dr. Robinson, who based his account on a "stuffed" specimen, supposed to have been shot about 1765

Jamaican/Gosse's Macaw†     Ara gossei
Rothschild  1905
Jamaica: Hanover Parish, Lucea.  Mountains
split from: Ara tricolor
insert after: Ara tricolor
Walter Rothschild,
On extinct and living parrots of the West Indies
Bulletin B.O.C. 16 (1905): 13-14
 

Not recognized as a separate species in HBW4.

Grey-breasted Parakeet     Pyrrhura griseipectus
Salvadori  1900
Brazil: Ceará, Alagoas, Pernambuco (Serra Negra)
split from: Pyrrhura leucotis
insert after: Pyrrhura leucotis
F. Olmos, P. Martuscelli & R. Silva e Silva,
The distribution and dry-season ecology of Pfrimer's Conure Pyrrhura pfrimeri, with a reappraisal of Brazilian Pyrrhura "leucotis"
Ornitologia Neotropical 8 (1997): 121-132
 

Not recognized as a full species either in HBW4.

Pfrimer's Parakeet     Pyrrhura pfrimeri
de Miranda-Ribeiro  1920
ce Brazil: Goiás, Serra Geral
split from: Pyrrhura leucotis
insert after: Pyrrhura leucotis
F. Olmos, P. Martuscelli & R. Silva e Silva,
The distribution and dry-season ecology of Pfrimer's Conure Pyrrhura pfrimeri, with a reappraisal of Brazilian Pyrrhura "leucotis"
Ornitologia Neotropical 8 (1997): 121-132
 

Not recognized as a separate species in Jiniper and Parr.

Cliff Parakeet     Myiopsitta luchsi
(Finsch)   1868
c Bolivia: s Cochabamba & adjacent Chuquisaca
split from: Myiopsitta monachus
insert after: Myipositta monachus
Nigel J. Collar,
in: Handbook of the Birds of the World. Vol. 4, J. del Hoyo, A. Elliott & J. Sargatal (eds.), Barcelona, 1997