I think parrots could potentially have evolved to be the dominant species on the planet if we hadn't beat them to it, although having to stand on their "hands" might have been a real handicap when it came to developing industry... ever read "West of Eden"?
On the other hand, in the current circumstance, they are not as intelligent as humans. Alex, who was a rocket scientist of an African Grey, was remarkable. He could count to 5, he had a concept of "zero" and he appeared to have a sense of humor. He was not, however, doing calculus, reading the newspaper or even counting to 10. So we are MUCH more intelligent on virtually every measurable scale. However, Alex is one case that is compelling in arguing that the difference between "lower animals" and humans is one of degree not kind. His level of intellectual function can be compared to a very young human child, at least in some respects.
Bird's brains are much, much smaller than ours are. That suggests that, while we use only a tiny fraction of our neurological capacity, they must be running close to full out to do what they are able to do. Which suggests their potential as they are right now, is much less than our potential should we somehow tap into all that unused grey matter.
I think one reason we are able to perceive parrots as being quite intelligent is that they are in several important ways similar to humans. That makes us able to appreciate what we are observing more than when we look at, say, a rat or a pig, which are also pretty intelligent but dissimilar to humans in those key ways.
I think that those key aspects are:
* They are very visually oriented -- they are driven more by what they see than what they hear or smell
* They are inherently oriented towards communicating using sounds, in the case of most parrots, adaptive/learned sounds. Most other animal species communicate much more frequently through body language or even smell.
* They have grasping feet that they use to manipulate objects around them in very much the same way we can use our hands, although with less finesse and only one at a time unless they are flat on their backs.
These things make it easier for us to perceive their intelligence, and at times makes them seem uncannily "human", but it is unclear to me how much more intelligent they actually are than other species. Cross species intelligence comparisons are very difficult because the social and survival skills needed vary so much from a cetacean to a parrot to a horse to a cat to a dog.
In addition, possibly as a result of what is needed to manage flight and complex social behaviors, they make correlations rapidly and easily. That is to say, they can connect cause and effect very efficiently, so they learn quickly, sometimes from a single repetition (hence they easily learn to scream and to bite if it gets them what they want!).
