There are no parachute units currently, and this is per design. Parachutes have pretty much been phased out on an operational and strategic level, tactically they are of course still used.
The only armed forces currently still maintaining a larger parachutist contingency is Russia and they too are more and more transitioning these troops into air-assault helicopter forces.
Reasons for this shift are numerous, but essentially date back to WW2 and the rather "flaky" results of large scale paradrops (Market-Garden/Arnheim, Crete to name two of the worst).
Additionally modern battlefields require copious amounts of munitions and supplies for frontline units to work effectively, with "self-sufficient" paratroopers being nearly impossible to maintain.
On top of this comes the high risk ratio of modern anti-air and air dominance scenarios - making air transporting large amounts of soldiers into contested air space more than hap-hazardous.

Hope I could explain some of our reasons why we decided to not include paratroopers in our game.

French army pretty much reservs paradrop (from high altitude) for its special forces, nowadays. Parachutes have a design problem : they have a high accident occurence, even with trained troops, and are counter intuitive with the modern doctrin of "control your space and your movement".

Parachutists die, and trained manpower is the most precious asset of an modern army.

I concur with Germanico on this one : as much swag a paratrooper will always be, they are not a "strategic tool" anymore.

In fact the paratroopers still "earn" their wings via parachuting but then use helicopters in combat.

One thing I would ask with what about paradrop able light armor. Some tanks like the Sheridan were designed to be dropped from cargo planes, onto the battlefield. The cargo plane would fly at low altitude, and have the tank deploy a cargo shoot, pulling it out of the plane. The crew would parachute from a different plane land near the tank, get in it, and drive to the battlefield. This for sure isn't paratroopers, but would be a cool feature, Im not sure if it other countries had vehicles like that or not. If not it could maybe be used as something specific to the Sheridan. Since the maneuver of dropping the tank could be risky, maybe it would have a chance of damaging your unit, making people think about using only when necessary such as quickly getting some armor to infantry under enemy attack or troops behind enemy lines. It would be a cool feature, but it would take some effort to get to work fully.

You can actually air-assault certain units - check out their stats and look for "air-mobile".

//G

adding to what Germanico said, many units get their air assault capabilities with an upgrade down the research tree line, some of those you can upgrade with air assault-ability include (but by no means limited to): CRV, Tank Destroyer, mobile AA, towed arty etc.... (provide you play one of the more recent games..)

oops, i forgot to add a part about things like the CRV being airmobile to my post. I knew those were air assualt able, but that it was still by helicopters (or at least thats the model for it) I was just going off the plane drop thing for the sheridan, but after further research that not many other AFVs could do this, so that due to balancing it wouldn't work well in the game.

Yeah - dropping tanks out of a flying plane didn't really work out to well...

Visualising why airdropping isn't that popular in the military: youtube.com/watch?v=pRhFRGzh0s0