Green cheek conures are nicknamed the Ice Cream Cone Birds because they love being held just like an ice cream cone. The bird shown here, Blackie, leans back & pulls up a foot & leans one wing over the top of the holding hand & goes to sleep.
Ryoko on the other hand, prefers to lean back pull up one foot while perched on my thumb and enjoys having the rest of my four fingers wrapped around her back. "This" is just a variation on the theme and the nickname Ice Cream Cone Bird is quite fitting.