It only looked THAT big and furry during that short "Petrie verrrrrrrrrrrrrry scary!" scene.
Yes, if you were to take your finger, and touch/pet Petrie's neck ring, yes it would feel soft.
Their neck rings aren't a "hard" part of their bodies, it is a little furry and soft to the touch, Petrie's, His mothers, brothers/sisters, Pterano's, Rinkus's, Sierra's... their neck rings would all feel soft to the touch.
(Even though you don't always see any "hairy/fur" detail.. They are actually soft.
Now Petrie's here in the picture, all inflated is OVER-exaggerated, thinking that he's really scary PUFFING out his neckring like that! LOL! ^_^

But take it from me, guys, a flyer's neck ring really is soft/fluffy to the touch, and it's one of my favorite things about them, too!
