Yes budgies are parakeets that are a type of parrot and cockatiels are parrots also.
I meant that undomesticated species of parrots that are commonly kept as pets; African grey parrots, caiques, quaker parrots, conures, macaws, cockatoos, ect.
As a bird nerd I’m gonna say:
Parrots aren’t domesticated, even the small ones. They are companion animals, but they’re not really that distinct from their wild populations. Certainly some of the smaller ones have been bred into a variety of different colours and appearances, but a fancy colour doesn’t necessarily mean they’re domestic. A British-style show budgie is toeing the line though…
Chickens and fancy pigeons are domesticated. They are very, very different from their wild progenitors. Argument might be made for some other gamefowl as well – geese, ducks, etc.
Chickens and pigeons are great bird pets. Quail are also good, at least the domesticated varieties- wild varieties are too flighty. They can be kept in relatively small enclosures, can be brought out for exercise with minimal risk of escape, and are happy with simple toys and food enrichment. They need entertainment and socialization, but aren’t comparable to parrots. Also, it’s easy enough to keep several of them so they socialize with each other. All three, especially chickens and pigeons, can be extremely loving with their owners and can learn tricks.
Finches aren’t really domesticated, but can also be good pets, since they’re tiny and not ultra-smart. They still need plenty of space to fly, they have to be kept in groups, and they’re more of a pretty pet than an interactive pet, but they can be kept without a massive lifestyle commitment.
And you know what else is great about those four bird groups? They can be bred by simply keeping them in normal, good conditions, offering nesting places, and feeding the right diet. No isolation or boredom required. You may have to take and incubate the eggs from some that aren’t good at brooding, but finches, pigeons, chickens, and quail will all breed at a fairly steady rate under normal conditions.