I have been out of the Army for 47 years so I have little knowledge of how things work now.
Back in my days Airborne training was a 3-week program. First week was ground week, lots of PT (physical training), running, and basic instruction on PLF (parachute landing falls). Week two was tower week, starting with jumping from moderate heights in full equipment to practice your PLF, then progressing to the big towers on which you were hauled up to a considerable height (don't remember the actual height, but it was at least 100 feet) in full harness and parachute supported in a halo-shaped device, then released to descend on the parachute to the ground, landing and recovering. Third week was jump week, a jump every day, supported by USAF aircraft (C130 and C119 cargo planes), starting with only the harness and parachute, then progressing to full equipment jumps (weapons and field gear, kit bags rigged to the harness). End of jump week was graduation day, awarding the "jump wings" to those who had successfully completed the program.
Basic "jump wings" (parachutist badge) was the standard of qualification for airborne duty and assignments. "Senior" parachutist badge and "jumpmaster" qualifications required advanced training (and as others have pointed out, not everyone automatically revolved into those schools, mostly career officers "getting their tickets punched"). Jump pay was $55.00 per month for enlisted soldiers, $110.00 per month for officers during my service. The soldier had to be assigned to an "airborne" unit and assignment, and complete at least one jump every 90 days in order to qualify for the added pay. Might not sound like a lot of money these days, but back in the day when a PFC was paid $150 per month, a corporal about $210, and a buck sergeant made $250 or so, the extra $55.00 per month was HUGE.
I have had the good fortune to know several old WW2 paratroopers, and have shared experiences with them. Back in the 1940's the airborne units were trained together for months, and the training went far beyond basic parachute school. They were extensively trained in small unit tactics, weapons (US and foreign), advanced first aid and medical care, survival, escape & evasion, hand-to-hand combat, intelligence gathering, camouflage, and many other aspects of the military "arts".
Comparing "airborne training" of the past 50 years or so to "paratrooper training" of World War II era is like comparing juicy ripe apples to shriveled grapes.