OK, lets look at the major paratrooper operations of the war (regiment and up), their objectives, how they were executed, and the extent of the ground the paratroopers took:
Netherlands, may 1940. Two-fold operation; targets: capturing airfields near The Hague (wiped out by security-level defenses), and capturing major river bridges for the advance of 9th Panzer division (succesful). Territorial: no more than 5 km diameter.
Crete, may 1941.Airborne troops dropping on one major and two minor airstrips. Reinforcements flown in by regular transport planes to those airfields after capture. Territorial by air drop: no more than 3x 5 km.
Vyazma, Russia, january 1942. Corps (planned) / Brigade (actual)-sized unit dropping in the rear of German army group Center to prevent their retreat. Unsuccesful, widely scattered. Linking up and incorporated into ground forces afterwards. Territorial gains: hard to find sources, but very limited.
Sicily, july 1943. British and American paratroopers taking and holding tactical objectives (road crossings, high grounds, and bridges), meant to support the advance of a seaborne invasion force. Territorial by air drop: several pockets each less than 5 km diameter (though the Americans were scattered over a much larger area, they operated mostly as commando/raid troops without holding ground).
Dniepr, Russia, september 1943. Russian air drop to reinforce an existing bridgehead across the Dniepr river in the aftermath of the Kursk battle. Territorial: 25km, together with ground troops.
Normandy, june 1944. Three American/British divisions landing at the outer borders of the intended beachhead, mainly to block German reinforcements deployment to the battlefield. Contact with seaborne invasion troops on the same day. Territorial by air drop: < 15 km (101st on Cherbourg peninsula).
Southern France, august 1944. 5,000 mixed troops dropping to secure advance routes for a seaborne invasion. Territorial: less than 10 km.
Netherlands, september 1944. Three American-British divisions plus a Polish brigade landing in a "string" to secure a highway and the bridges in it, to ease an advance of British ground forces. Territorial: about 50 km long, 3-10 km wide, with many gaps in between (that's still much less than a CoW province!)
Germany, march 1945. Two British-American division dropping across the Rhine river to secure exploit routes for ground troops crossing the Rhine by boat. Territorial: about 15km diameter.
CONCLUSION:
With the possible exception of Crete, ALL of those operations were in direct conjunction with a regular ground attack or seaborne invasion. Closer examination of each of those operation will show that the paratroopers would have been quickly and completely wiped out (indeed, none of those operations would ever have been executed), if no ground/seaborne troops had been involved. Paradrop ONLY conquest of a CoW-sized province (and please don't start about those small island provs now)
was done by air drop only ONCE (Crete), and even then, only when additional ground forces were flown in by transport planes.