Expanded from an earlier answer of mine

The Germans certainly did manage to convince much of the world of the validity of airborne warfare. The airborne assaults in Scandinavia and the Low Countries in 1940 were spectacular successes and Crete was on the surface a continuation of these string of victories. But Crete was the high-water mark for the Fallschirmjäger. There were some minor airborne operations in 1942 and some Fallschirmjäger troops were airlifted around to various sectors, but there never were the type of large-scale combat drops of the early war period. This had both to do with a general skepticism of the efficacy of airborne warfare as well as a long-term process that destroyed the Luftwaffe's transport arm.

Barbarossa was unlike earlier German campaigns in the war in that there were no major air landings or airdrops. There were some very small-scale airdrops during the opening phases of Barbarossa, but these were nothing like the Fallschirmjäger operations in Scandinavia, the Low Countries, and the Balkans. The VII Flieger-Divison and some Fallschirmjäger regiments were also employed on the Leningrad front late in the campaign, but they operated as a ground-only units. This distinct lack of paratroops in the campaign can be traced to two interrelated causes.

The first and most obvious reason was the massive losses the Fallschirmjäger incurred in the invasion of Crete. Although successful, Merkur ripped the guts out of Kurt Student's expanded parachute corps. The invasion exposed real problems in German parachute doctrine, equipment, and tactics. Earlier airdrops also had high losses, but the victory at Crete was Pyrrhic. Much of the Fallschirmjäger units had to spend most of Barbaorssa reconstituting themselves. Additionally, these losses made German planners increasingly doubtful about the efficacy of mass airborne assaults. Hitler had conceived of using an airborne assault as a component of 1942's drive into the Caucasus, but German planners thought it not worth the effort. There was some headway made in planning a redux of Crete over Malta, but events on the fronts rendered the planning for Operation Herkules moot.

The other reason for the lack of Fallschirmjäger airborne operations in the USSR was the Germans were starting to come up quite short on transport aircraft. Here Merkur too extracted a high price on Ju-52s and the Luftwaffe's transport arm had been hemorrhaging pilots and planes from the start of the war. These losses were beginning to outstrip replacements as early as 1941. To make matters worse, the latter stages of Barbarossa became more reliant on airdropped supplies as the Axis forces overstretched their logistical networks. Resupply by air became one of the means that isolated German units such as those in the Demyansk Pocket were able to fight on in through the winter campaign. The Luftwaffe kept up this logistical resupply by stripping training units of both aircraft and their instructors. This may have worked over the short-term, but this fatally damaged the long-term viability of the transport arm. Things were bad for the transport arm in 1942 and the Stalingrad airlift made matters exponentially worse. The losses both supplying Tunisia and Stalingrad removed well over a third of all German transport aircraft. Added to this problem, the Luftwaffe's training regimen was not only ill-prepared for replacing pilots, but German industry could neither produce an aircraft to replace the Ju-52 or make good the horrendous losses suffered on top of Merker, Stalingrad, and Tunisia.

This lack of aircraft meant that even when the Fallschirmjäger made rebuilt their establishment from Crete, they lacked the aircraft to drop them. A starker contrast could not be made with the Anglo-American forces. While there was considerable friction over the parceling out of transport aircraft, the Anglo-American airborne troops had an embarrassment of riches compared to their German counterparts. The Germans simply did not have the aircraft or the fuel to expend on mass airborne drops by the end of 1942 while the Western Allies increasingly had the planes and the fuel to drop them en masse. Anglo-American airborne doctrine was also slightly better thought out than the German one as it was more closely related to combined operations with ground force. Furthermore, there were the Anglo-Americans had developed slightly better parachute equipment, were not as dependent on airborne canisters for heavy arms and ammunition, and favored night drops. These all worked to lessen the losses in Allied airborne drops so that while Anglo-American paratroopers suffered high casualties, they did not become Allied versions of Crete. The reduction of the Fallschirmjäger into ground-based infantry that only occasionally made small-scale combat drops was a reflection of the brittle nature of Germany's war effort in which attrition rendered a once revolutionary force into troops to fed into emergency situations.

Sources

Center of Military History. Airborne Operations: A German Appraisal. Washington, D.C.: Center of Military History, U.S. Army, 1989.

Citino, Robert M. Death of the Wehrmacht: The German Campaigns of 1942. Lawrence: University Pr Of Kansas, 2011.

Militärgeschichtliches Forschungsamt. Germany and the Second World War: Volume VI. The Global War. Oxford: Clarendon Press, 2001.