Back to question in original post: the answer lies in the concept of "home". I think for most parrots, flock is home. A parrot never stay too far away from its flock. If there is a secret of free flying parrots, this is it.
I don't train my birds nowadays, but I have a flock. I am able to take my cockatiel outside without harness if I can be bothered following him while he learns so that the flock stays together. (NOTE: Tiels tends to wander their own way so please don't take yours outside if you or other flock member(s) can't stay with it all the time. They have wings and even a cage potato can still fly a mile before fatigue. Same with any parrot without outdoor flying skills really!) I also have my conure wandering the neighbourhood as he pleases whenever I am outside doing things. He occasionally comes for walks or car rides with me, never restrained.
And yes human can become a flock member. you need to be an important/valuable member of the flock if you would like the birds to stay around you.
Also on the side note, being aware of flighted predator is an instinct. No training can ever get rid of it. A parrot can be slow and a poor flyer, but they do know when they see a hawk or gull. A Human predator is another story if they are too tame :p