The eclectus's at the bird store LOVED to cuddle with me, but it was on their terms. Sometimes, they were very reluctant to have their feathers touched, but they seemed to be a calm and serene type of bird.
My caqiues are extremely cuddly, especially before bed or in the morning after they wake up. They love to be rubbed behind their head, chest, and will even roll over like a dog so you can scratch their tummy. Of course, when they are active and crazy, they usually dont like long periods of cuddling.
Honestly, I can't think of a bird that would not cuddle with you as long as they are comfortable with you. I have seen many youtube videos of cuddly macaws, cockatoos, green cheek conures, sun conures, etc... I dont think you can go wrong
I had the same question as you. I was getting a GCC, I wanted a cuddly, entertaining, Snuggling bird. She said GCC's can be TAUGHT all of this, since they are a bit jumpier, nippier, But with a Su Conure it is NATURAL. They love attention. Which is why they scream. Because they want attention. I wanted a snuggly bird, and that's what I'm getting A Sun Conure
Cockatoos are famous for cuddling, as well as screaming their heads off and biting hard. My Amazon cuddles sometimes. She likes getting head scratches but prefers to remain perched while she gets them. If it gets any cuddlier than that (like if I pet beyond her head too much, or pet her late at night) she thinks its mating time and starts getting a little too sexual. She loves attention but generally prefers the verbal kind to the tactile. Whatever kind of bird you get, though, it's going to take patience and a build-up of trust before too much cuddling happens. It's not like having a dog, where you can pet them at any time.
Cuddly in a parrot, I think, is far more dependent on the personality of the bird than the species. Amazons, Macaws, Pois, several others, but if the personality of a particular bird isn't interested in cuddling, there is nothing you can do.
I have a feisty green cheek that doesn't like to be cuddled, but most conures I have heard are cuddly. Painted Conures, Jenday Conures, Suns, Green Cheeks... They're all great. It really comes down to how loud you can handle and price. Green cheeks are quiet and pretty cheap, Painted Conures are quiet and more rare so more expensive, and Suns and Jendays are loud but cuddly. They're also more expensive.