Could the same be applied to para training?
No. The PJIs do more than just training - they also oversee all aspects of the operational jump from within the aircraft, performing safety checks and coordinating the sticks (when jumping from both doors). They could not be civilian, unless you want to send civilians into the battlespace.
I'd agree with Airborne Aircrew that parachute training should remain with the RAF. Paras are just one client - albeit the largest. A typical para course will be about 80% para, with the rest made up of RM, a few gunners and medics (after P-company, so wearing maroon berets), a few baby SF, RAFR, and even the occasional fishhead (SPAG types). Best to have that lot controlled and trained by a neutral cadre of RAF. Besides, as AA also pointed out, the aircraft are controlled by RAF so RAF PJIs fit better into that command structure.