actually it's neither of those: http://commons.wikimedia.org/wiki/File:US_Army_Airborne_basic_parachutist_badge.gif
They are talking about the slim pin, not the airborn pin. Also its not an airborn pin Clay wears, there is a skull where the parachute would be.
The pin with the skull instead of parachute is a COC (coalition of clubs) pin
Clay has two pins: one is very slender so it's easy to mistake it (I thought it was a harp too, at first), but it's part of a hooded reaper holding a scythe. I thought the second was the basic Parachutist Badge (Jump Wings), but I got a closer look and noticed it was a Winged Skull, which looks identical to the unofficial pin for the MACV-SOG Paratroopers, a covert, special forces group.
I posted more info about it in a General Discssion thread. Link: