Dogs and cats occupy a unique place in the human psyche. They self domesticated to become our companion species by being both useful and affectionate towards us. We have cultural and biological reasons to feel differently about them.
ETA: or, what everybody else said. 
Well most people aren’t either.
I believe that cats and dogs trigger the parental instinct in each of us. Many people love to take care of and cuddle cats and dogs because they have an instinct to take care of and cuddle children, and cats and dogs are sufficiently close to children in terms of stimuli that the area of the brain responsible for that instinct fires and releases endorphins (pleasure chemicals).
Parrots are extremely intelligent, way more so than dogs and cats. Some of them, like the hyacinth macaw, have been shown to have as much intelligence as a small child. Yet no one cares about them like they do cats and dogs.
It’s entirely social and inherent in our culture.
Also note that puppies and kittens have big heads and big eyes (in proportion to total body size), like human babies. This is something we’re genetically programmed to respond to, and something that is lacking or hard to recognize in most other pet species.
Well, parrot lovers care about parrots… it’s just that most people don’t really know about or understand pet birds. Certainly, communication with birds, even intelligent birds, is not as intuitive as communication with dogs or cats. Birds are not mammals, they really are more different from us than other mammals.
That said, there is concern about overbreeding pet birds, but most people seeking pet birds aren’t seeking the highly bred “show” birds in the same proportions as dog people seeking a purebred. Another factor is that birds have not been domesticated (actually, it’s more that each generation is tamed anew - parrots aren’t “domestic” the way dogs, cats, cattle, etc. are) or artificially bred for nearly as long as dogs and cats (with pigeons being an exception). Most parrot species have been bred in captivity for pets for maybe 100 years at most - and with lifespans of some of them running to 50 years that’s not a lot of generations. There hasn’t been as much time for overbreeding and the genetic damage that it can bring.
What breaks my heart with pet birds is that there are so many people who just lock them in a cage as pretty objects - birds of the species kept for pets are highly social animals that need a lot of social interaction or they suffer terribly