Parrots do not socialize with non-parrot species. If you're keeping a mixed flock that includes something other than parrots, you will need a big enclosure. Even then, you will have to make sure there isn't a big size problem - a large parrot can easily kill a smaller bird (including smaller parrot), so never mix your bird species without asking for expert advice first.
The main problem with parrots is their hooked beaks. These can hurt or kill other birds, and even a minor squabble can result in Canaries and Finches with snapped legs, nasty cuts, or worse.
These Cockatiels have settled into a mixed aviary - but you'll need something at least this big for it to work
As a rule of thumb, don't mix anything larger than a cockatiel with other types of bird. Lovebirds and Lories may look small and cute, but they can cause havoc in a flock of finches. It's worth saying again: parrots and finches will not mix successfully unless you get the size-matching right, and unless you have a big enclosure for them.
Keeping Parrots with Quail
Conures, Cockatiels and Budgerigars can be kept in the same aviary as the species of pet Quail available in the pet trade (usually Button quail (aka King or Chinese quail) (Coturnix chinensii), and Japanese quail (Coturnix japonica). The parrots won't take any notice of the plump little Quail zipping around on the cage bottom like wind-up toys, and the Quail will ignore the parrots too. However, larger parrot species will try to investigate the Quail, and the quail will panic when that happens. It's best not to keep big parrots and quail together, for this reason. Some of the docile Amazon Parrot species are exceptions to this general rule.
Small breeds like this Bourkes parakeet can cohabit with quail
Keeping Parrots with Finches and Canaries
Finches can mix with Conures, Cockatiels and Budgies, but not with any species larger than that - and only in a very big aviary. Canaries are not very good mixers, however (and the same applies to other timid finch species), and in general they will not be happy in a mixed aviary that includes parrots, of any size.
Budgies and Zebra Finches are neighbors in their native Australia
Keeping Parrots with Chickens
This is a simple one: don't! Chickens eat anything and everything, they are naturally aggressive to anything smaller than them, and will not hesitate to attack smaller parrots if they're in reach. Larger parrots will be able to fend off the chickens... but that's not exactly a match made in heaven either. So, on the Chickens and Parrots question, it's a definite Cock-a-doodle-don't!
Keeping Parrots with Doves
Doves like to keep things calm and quiet. They are only happy if surrounded by like-minded species. That rules out most of the parrots! However, Conures and Budgies can be kept with aviary doves successfully. We're not talking about any old town pigeon, but the dove species sold in the pet trade. Common pet dove species include the Ring-necked dove (Streptopelia capicola) and the Diamond dove (Geopelia cuneata).