Yes, there were numerous smaller airborne operations in World War 2 on both the Axis and the Allied side. I can list a few notable ones, but there are many worth checking out.

One great example is Operation Biting (a.k.a. the Bruneval Raid), a British airborne raid on a suspected German radar technology installation in northern France in late February 1942. RAF aerial reconnaissance had spotted several sites the purpose of which they could not determine, but they suspected they were related to the Germans' very vicious defense of occupied Europe against Allied bombing. The British researchers wanted the raid to raid one of the installations, disassemble what they found, and bring it back for study. The troops were dropped in, neutralized the defenders, disassembled (most of) the Würzburg radar they found, got to the beach, and were successfully extracted without major incident. Also of note is that John Pohe was a pilot on this mission. You can read more about his life here.

The first British airborne operation was Operation Colossus, a year earlier in February 1941. This small raid aimed to disable an aqueduct near Calitri in southern Italy. Like most airborne operations, it did not go according to plan, but the objective was accomplished. Most of the paratroopers were captured and became POWs. The aqueduct was swiftly repaired and there was little strategic damage done. Still, it was the first effort by the British in airborne warfare and they would apply the lessons learned to future operations, like Operation Biting.

On the Axis side, the German airborne invasion of Crete is usually the first to be mentioned. But the Germans did smaller raids, as well. Notable is their attempt to seize the Corinth Canal bridge on 26 April 1941 (their "Operation Hannibal") and prevent it from being destroyed by the retreating Allies. Due to (reportedly) some German engineering team blunders, the bridge was destroyed despite the paratroopers successfully having removed the Allied explosive charges on the bridge.

Others to check out:

• The Battle of Elephant Point, performed by a Gurkha airborne unit in Burma in May 1945

• The Raid on Los Baños in the Philippines on 23 February 1945, which was a rescue operation freeing over 2000 allied prisoners from their Japanese captors

• the assault on the Belgian Fort Eben-Emael by German Fallschirmjägers in 1940

• for an example of a Japanese airborne operation, check out the battle at Palembang, where (among other things) Teishin Shudan jumped on the airfield

And these are just a few examples among many. All of the above are well worth reading about in detail. For American airborne operations, see here. For German appraisal of their airborne methods, see here. Here is a generalist but well-written overview of the topic.