I think the best classification of Psittaciformes is this one:
I like your taxonomy, it looks as mine, but I include fossil taxa:
Fossil taxa in red
Ordo Psittaciformes Wagler, 1830;
Subordo
Vastanavi
Familia
Vastanavidae
G. Mayr, Rana, Rose, Sahni, Kumar, Sing et T. Smith, 2010
Subordo
Halcyornithi
Parvordo
Halcyornithida
Familia
Halcyornithidae Harrison et Walker, 1972
Parvordo
Quercypsittida
Familia
Quercypsittidae Mourer-Chauviré, 1992
Parvordo
Namapsittida
Familia
Namapsittidae Mourer-Chaviré, Pickford et Senut, 2017
Parvordo
Mopsittida
Familia
Mopsittidae
Subordo Psittaci Wagler, 1830.
Superfamilia Strigopoidea C. L. J. L. Bonaparte, 1849
Familia Incertae Sedis
Familia Strigopidae C. L. J. L. Bonaparte, 1849
Familia Nestoridae C. L. J. L. Bonaparte, 1849
Superfamilia Cacatuoidea G. R. Gray, 1840
Familia Cacatuidae G. R. Gray, 1840
Subfamilia Nymphicinae
C. L. J. L. Bonaparte, 1857
Subfamilia Calyptorhynchinae
C. L. J. L. Bonaparte, 1853
Subfamilia Cacatuinae G. R. Gray, 1840
Tribus Microglossini
C. L. J. L. Bonaparte, 1853
Tribus Cacatuini
C. L. J. L. Bonaparte, 1853
Superfamilia Psittacoidea Rafinesque-Schmaltz, 1815
Familia Psittacidae Rafinesque, 1815
Subfamilia Psittacinae Rafinesque, 1815
Subfamilia Arinae G. R. Gray, 1840
Tribus Amoropsittacini
Brereton, 1963
Tribus Brotogerini Wolters, 1975
Tribus Androglossini
Sundevall, 1872
Tribus Arini G. R. Gray, 1840
Familia Psittaculidae Vigors, 1825
Subfamilia Psittrichasinae
von Boetticher, 1959
Subfamilia Coracopseinae
Joseph, Toon, Schritzinger, Wright et Schodde, 2012
Subfamilia Psittaculinae Vigors, 1825
Tribus Incertae Sedis
Tribus Micropsittini
Mathews, 1927
Tribus Polytelini Mathews, 1916
Tribus Psittaculini Vigors, 1825
Tribus Psittacellini Wolters, 1975
Tribus Pezoporini
C. L. J. L. Bonaparte, 1838
Tribus Platycercini Selby, 1836
Tribus Agapornini Salvin, 1882
Tribus Loriini Selby, 1836
Superordo Psittacopedimorphae Superordo Novum
Familia Psittacopedidae G. Mayr, 2015
Fred