The rule has to do with the CHL. Zach Parise as an NCAA player has no age restrictions on playing in the AHL. If you play a year in the CHL then you have to either have 4 years of CHL experience to play in the AHL or be 20 years old.
Same situation for Ryan Kesler of Vancouver this past year.
Edit, I didn't see the Stortini post. If he has played 4 years of junior and is 19 or is 20 years old he can play in the AHL.
My other guess is that it could be an amateur tryout contract. The Penguins just did the same with Ryan Whitney.