That's easy enough – preowned parrots!
You're usually stuck with no history, no clue what the first person or persons did right wrong or indifferent with the bird, and some less reputable people will gloss over known problems with the bird just to secure a new home for the bird.
There are so many damaged psyches out there, and you're going to have to decode every single element that the bird is going to try to communicate as to what went wrong so you can make it right.
Statistically speaking, the dumping grounds are usually filled with cockatoos, macaws, timeh African grays(to a much lesser extent Congo African grays, although still represented numbers), Amazons, cockatiels, parakeets, and the occasional eclectis or pionus.
I can explain the cockatoo situation, but somebody already did it better
All About Cockatoos - MyToos.com
That website should be mandatory reading for anyone considering a cockatoo. Misunderstand me not, I even have an actual favorite breed of cockatoo {bare eyed cockatoos}, and I would gladly welcome one of those to my home. The reason I would do so however, is because I'm educated about the species. Deliberately ignorant bird owners cost birds homes :-(
As to the macaws, the actual behaviors specific to the species and then subdivided by subspecies and hybrids – well, it's a hot mess to put it mildly. Getting any information at all that's actually useful has to be compiled from message boards like this one and weighed against multiple postings of the exact same behavior in order to decide whether or not it's typical. I actually became frustrated enough as a new macaw owner to compile my own list. It's still a work in progress, but the fact remains that I had to do it means that no one else did it or else I would've been able to find it all in one place.
In any case, much like the cockatoos, macaws are extraordinarily emotionally high maintenance birds. You get a little more wiggle room for major long-term bird interaction mistakes with macaws, but not by much if you don't make corrections.
I really can't figure out what the hell is going on with the African grey owners out there. I'm referring to the irresponsible idiots I've come in person contact with since my own bird was murdered. Their excuses often express knowing the exact right thing to do, and then not doing it--WTF!
Amazons are subject to lack of full disclosure, which is a major problem with a lot of parrot species at time of original sale. Because when they're overly excited, they can become hormonal and their ability to reason goes to hell for a few minutes, the idiot that just overexcited the bird has no idea why their fingers are bloody and the bird( whose since calmed down) has no idea why the human is mad at it. That's the worst case scenario. Mostly people are just treating cheaper birds disposable, which brings me to.....
...... wait for it.....
Cockatiels and budgies
Other than allergies(something to consider with cockatiels), there is absolutely no other rational reason someone should be dumping these birds. There considered so disposable that they don't even require a license to own in the US, despite almost every other parrot species requiring one.
And let's not forget the other factor involved – time – as in the human has/makes less time for the bird, which can also lead to disastrous emotional instability to the point of self-mutilation.
There is a reason of a lot of the birds in rescues are plucked nearly naked, and it's not because they're looking for a career in softcore porn.......
As somebody already mentioned, if you're going to go species specific,Keas-- and add most of the cockatoos, all of the Vassas, those tiny bee winged whatever you call them(can't even keep them in a regular cage, but require a large flight aviary), and that's not a complete list – just off the top my head.