The study was done on primates with a view to apply it to humans (who are primates, too) but it can be easily extrapolated to parrots. It states that the two different theories to explain social groups (paternal care and female spacing) are actually complementary and not opposed. What I liked best was the assertion that pair-bonding is a step to a more complex society - which makes sense taking into consideration parrots intelligence.
https://www.zmescience.com/science/pair ... -89235234/
PS I believe that parrots do much better if they have a mate or, at least, a companion of their own species.