Although I have had a few birds during my life. I was fortunate enough to have my dream bird. His name was Marlin and he was an Alexandrine parakeet. Marlin was with us for two and half years before he was spooked and flew off. This was on the 8th of August 2015. I don't believe that the species should be the dream, but more the connection between you and your bird. Marlin was my dream bird not because he was an Alexandrine, it was because he was companion bird to the family and became a family member, in all context of that statement. I'm glad that we had time with Marlin and his absence has left a void in our life and we still miss him terribly.
Like most parrots, they are flock orientated and will interact with their human flock. Marlin was very social and preferred to be with us than his toys. He was not hard to train. The secret is time, lots and lots of time of bonding and training. Like all animals if you want a well-behaved and trained animal then you need to put in the time. This is my personal experience. Alexandrine Parakeets are known has gentle giants and are up there with the African Grey and Eclectus for intelligence. Alexandrines are AMAZING bird and don't believe everything you're told about Alexandrines.
Marlin was quite happy to sit on any of the family's shoulder and chat to them. When Marlin was young the family would sit down in a circle and play pass the parcel with Marlin. This meant that each family member would get Marlin to step up and then he would get a scratch and a rub, then pass onto the next family member. This was done until Marlin was able to fly around the house and he would always fly onto someone shoulder have a chat and move on.