Yeah, parrots are very smart and have very complex social structures. The studies by Irene Pepperburg and her African Grey Alex were at one time groundbreaking because if you dissect a parrot brain, it looks very simple and it looks purely limbic but it's actually not. They actually have a very complex structure similiar to our cortex but it's just arranged a bit differently. For years, scientists dismissed any signs of avian intelligence as us anthropomorphizing because they looked at their brain as simple. I think it was partly Alex and partly the fact that we understand human brains better now that made science dig further into them.
Birds actually have a complex language, accents (they can recognize where other birds are from), social hierarchies, etc. They learn to live in our human world and can even compose sentences, words, etc. to prove that "understand" human language (like a child does). I'm not saying a parrot will ever have an intelligent conversation with anyone (although considering some of the humans I talk to, I'd rather talk to a parrot) but they can learn the same basics that human babies learn. They can learn to categorize (which is one of the first steps of human intelligence) like items, different items (they can tell you what the difference is "size" "shape" "color"), can compare items in a category ( "bigger" "smaller") and have a sense of self. All of those are things used in human babies as signs of intellect development. The first thing kids learn is to categorize like items and then they learn differences and self-awareness is usually one of the later sign of intellect.
Of course, like a kid, a parrot has to be taught. Not all parrots are geniuses either. There are "slow" ones just like in humans. I swear mine is a moron but one at the zoo, I bet if we tried to train him he'd be a real genius. You can just tell he knows stuff and he takes it in. He, like a baby, imitates human actions and sounds, trying to put it together. It's hard to be a bird operating in a human world though. My dumb bird just hangs upside down, screams hello and makes goofy eyes like "duh" all the time, poor little moron.
Oh yeah, ravens are smart too and other corvids. They've been known to make tools and to "learn" how to make humans do their work for them. Suprisingly enough, vultures (new world) are actually pretty intelligent too. Probably below parrots and corvids but they rank pretty highly compared to some other birds.
I could go on but, I don't know what this has to do with rifftrax