Are they fully reinforced?

yes they are fully reinforced. I tested by starting the 39 scenario as Axis, building 2 x Airbourne Corps (1 x Level 1 and 1 x level 2) and then moving them each adjacent to an enemy unit. If you create the red attack arrow and then look at the bottom left menu you will see that they are reported as having 0.00 attack strength. If you carry out the attack the Airbourne Corps will lose strength and the enemy unit will remain undamaged. This is as the attack odds are less than 1:1. Do the same thing with a 4 strength infantry Corps and you will get an entirely different result.

Happens to me, too, but I figured that was because their supply route is usually cut off after they have parachuted (however, that would more or less defeat the very purpose of having them in game, apart from taking undefended enemy cities for a short time).