Out of all the birds we have, I think our parrotlet and cockatiel are the most cuddly, but with that being said, both our sun and green cheek conures are very cuddly, so is our DYH Amazon. We also have a quaker that loves to cuddle too.
The smaller birds like to cuddle in our hair while perching on our shoulder but Koolaid, our sun conure loves to go under the covers, cling to a shirt, and go to sleep while I watch tv.
A lot of it depends on the individual bird probably more than even the species of parrot, but in general, from my personal experience I'd say if you want a seriously cuddly bird, the parrotlets, conures, or cockatiels won't disapoint you.