The minimum recommended cage size is 1.5 to 2 times your parrots adult wingspan! CLICK to find yours

RESEARCH.... Amazing actual study info from renowned Aviculturalist Tony Silva!

" I have visited breeding facilities throughout the world and have 40 years of experience in breeding parrots; to date I have bred 82% of the approximately 350 parrot species. I have also studied parrots throughout the world and have amassed significant data on breeding biology, diet and behavior. This experience and data has led me to come to some conclusions.

The perception of cage size requirements has evolved over decades. Thirty years ago I bred African Greys Psittacus erithacus in cages as small as 90 cm (3 ft) square, feeling that the small enclosure met the requirements for security that these parrots needed. Over time fertility and reproduction began to wane. This species was at the time readily available as imports and the pairs were replaced. But my curious mind thought of an experiment involving Maroon-bellied Conures Pyrrhura frontalis to prove whether the waning reproduction in the African Greys was due to cage size or some other factor. For the trials 12 pairs of Maroon-bellied Conures were acquired. Six pairs were placed in cages 60 cm (2 ft) square and another six were placed in flights 1.8 m (6 ft) long x 60 cm (2 ft) wide x 90 cm (3 ft) high. Within 14 months all of the birds, which were fresh imports, began to breed. After a period averaging 7 years the pairs in the 60 cm square cages became woefully unproductive, with many clear eggs, while those in the 1.8 meter (& ft) long enclosures continued to breed successfully. When several of the pairs housed in the small cages were placed in the 1.8 m (6 ft) cages their fecundity returned. The diet, lighting and room temperature was identical in both cases. This suggested that the inability to fly had long term effects. The pairs in the 1.8 m long cages continued to reproduce for another 5 years, when the experiment stopped; the space was needed for another species. These results clearly demonstrated that enclosure size directly affected long term reproductive health.
My minimum enclosure recommendations follow; if the birds can be given more space than by all means it should be offered. All of the enclosures are suspended, so as to reduce the birds coming in contact with the ground, where feces, spilled food and a plethora of parasites can congregate. These cages also make vermin control (including rodents and snakes) easier when combined with a small mesh size; I prefer a mesh of 13 x 75 mm (1/2 x 3 in), which can exclude most snakes and all but the smallest rodents. The floor of the suspended cages should be pressure washed regularly to maintain hygiene.
Large Amazons, large macaws, Hawk-headed Parrots Deroptyus accipitrinus, African Greys, Indonesian and Australian parakeets (excluding Neophema), Asiatic parrots, Eclectus: 3.6 m (12 ft) long x 1.2 m (4 ft) square.
Small Amazons, large conures (i.e., Patagonian Cyanoliseus patagonus), large Poicephalus, large lories, fig parrots, miniature macaws, Neophema parakeets: 2.4 m (8 ft) long x 90 cm (3 ft) wide x 1.2 m (4 ft) high.
Cockatiels Nymphycus hollandicus, Caiques, small conures, Brotogeris parakeets, small lories, small African Poicephalus parrots: 1.8 m (6 ft) x 90 cm (3 ft) wide x 1.2 m (high).
Lovebirds, parrotlets: 1.2 m (4 ft) long x 90 cm (3 ft) x 1.2 m (4 ft) high.
For species that readily breed in colonies (some conures, Brotogeris parakeets, etc), the cage should be sized accordingly."

I think we can ascertain that larger cages = better health.  For both breeding and pet birds.