First jumps are from a tethered balloon on a fixed line, then they move onto light aircraft, mostly the islander series before moving onto the Hercules doing both side door and rear ramp fixed line jumps.
As others have said it is the Army that trains Para's, the RAF fly the planes for them to jump from.
Helicopter jumps are only carried out by specialist's as it takes a lot of training due to the extra complications caused by the Rotor downwash.
Also para's are only trained for fixed line jumps from standard altitude, special forces are trained in free fall and HALO (HIGH ALTITUDE LOW OPENING) Jumps