Parrots and parakeet type birds are territorial and aggressive birds, and they will fight for their own space. If you should put a Budgie and a Lovebird together the feathers will literally fly. They are just not compatible.
Cockatiel and budgie fighting as a demonstration of possible fight that might happen.
Finches and doves on the other hand are mostly not aggressive or territorial and you would be able to house some of the species together. Though even in these sweet looking species you will still find some kids of finches that are not compatible with each other, for instance a Java Finch will bully another finch with a less aggressive demeanour like a Star finch.
Rather keep the parrots apart, it will only hurt them and your heart.
