We often get asked whether the Night Parrot is related to other parrot species that are similarly green with yellow and black 'barred' patterning such as the Budgerigar, Kakapo, the Eastern and Western ground parrots and the tiger parrots of Papua New Guinea.
Well, to greater and lesser extents – it's complicated. The Night Parrot (Pezoporus occidentalis) is closely related to the Eastern Ground Parrot (Pezoporus wallicus) and Western Ground Parrot (Pezoporus flaviventris). They are, after all, in the in the same genus, Pezoporus.
Next closest are Bourke's Parrot (Neopsephotus bourkii) and the grass parrots of the genus Neophema (the Blue-winged, Elegant, Orange-bellied, Rock, Turquoise and Scarlet-chested parrots).
The Budgerigar (Melopsittacus undulatus) is more distantly related, being grouped with the lories and lorikeets.
These are all within the Psittaculidae family of parrots.
The tiger parrots (genus Psittacella) are also in the Psittaculidae but in their own subfamily.
And the Kakapo (Strigops habroptilus) is a distant relation, as far removed as it is possible to be while still being a parrot. It's in a separate superfamily – the Strigopoidea – along with the other New Zealand oddballs the Kea (Nestor notabilis) and the Kakas (Nestor meridionalis meridionalis and Nestor meridionalis septentrionalis).
The degree to which species are related depends on the length of time since they diverged from their common ancestor species.
All birds belong to the taxonomic class 'Aves' which appeared 155 million years ago, evolving from a therapod dinosaur ancestor.
Genetics analyses show that all parrot species (constituting the taxonomic order Psittaciformes) evolved from an ancestor species that split off from the Galliformes (fowls) sometime before 80 million years ago.
There are now 393 or so living parrot species. Within Psittaciformes there are three superfamilies, the Psittacoidea ('true' parrots), the Cacatuoidea (cockatoos) and the Strigopoidea (New Zealand parrots). These superfamilies separated from each other between 60 and 80 million years ago (see reference  below). The Night Parrot's Psittaculidae family is one of the three families within the Psittacoidea superfamily.
The dinosaurs went extinct 66 Mya and Australia separated from Antarctica about 36 Mya.
The Night Parrot diverged from the two ground parrots 3.3 million years ago .
And the Eastern Ground Parrot and Western Ground Parrot diverged from a common ancestor about 2 million years ago .
Some research papers with details about parrot evolution and taxonomy
 Tavares E.S. et al. (2006)
'Phylogenetic relationships and historical biogeography of neotropical parrots (Psittaciformes: Psittacidae: Arini) inferred from mitochondrial and nuclear DNA sequences.'
Systematic Biology, 55(3)
 Joseph L. et al. (2011)
'Molecular systematics of two enigmatic genera Psittacella and Pezoporus illuminate the ecological radiation of Australo-Papuan parrots (Aves: Psittaciformes)'
Molecular Phylogenetics and Evolution, 59(3)
 Murphy S.A. et al.(2010)
'A cryptic and critically endangered species revealed by mitochondrial DNA analyses: the Western Ground Parrot'
Conservation Genetics, 12(2)