There were multiple problems with using paratroopers for an invasion of the UK. The biggest was that the Germans never managed to achieve air superiority over the South Coast of England. The RAF was always able to put fighters into the air and engage German bombing raids. Paratrooper assaults require air superiority. They rely on slow, poorly armoured transports, which have to fly in a straight line, making them exceptionally easy targets for attacking fighters. Any attempt to invade the UK with paratroopers without the RAF first being suppressed would have resulted in heavy casualties to the attacking force before it could even land. The Germans had lost a lot of their transport aircraft during their operations in Norway, Belgium and Holland; as a result, they did not have sufficient aircraft to transport a major force to the UK. Secondly, paratroopers can only use what they can parachute with. This means that they do not have artillery, tanks or anti-tank guns. As a result, they are highly disadvantaged in combat against non-paratrooper units. While much of the British Army's equipment had been left on the wrong side of the Channel following the Battle of France, they still had hundreds of tanks and artillery pieces with which they could overrun any paratrooper beachhead. Finally, paratroopers, after they land, are all but cut-off from supply. They can only receive what can be flown or dropped in to them until they are relieved by ground troops. This means that they pose little attacking threat - they can drop on a position and hold it, but cannot advance out of the pocket without a better source of supply. The last of these problems could have been solved by an amphibious assault. Unfortunately, the strength of the Royal Navy made this a non-starter. For more information on this, see my answers here, here, here and here.