The info on PJI qual badges is here
Royal Air Force - Parachute Jump Instructor
In short, they originally wore an insignia on their sleeve, but by 1945 this had turned into a Brevet as they had also been doing the aircrew "air dispatcher" role.
Not so sure they do this these days though as an WSOp(ALM) is carried to do this, so maybe they should go back to the original sleeve insignia with more history behind it?
LJ