Re-enlisting?

Your Career Counselor is correct. You are obligated to three years at an Airborne assignment. When you accepted your current Airborne assignment, you agreed to three years on airborne. If you terminate your airborne status without fulfilling your obligation you will be subject to field grade UCMJ. Then you wouldn't be eligible to reenlist, and it would be a moot point.
The only airborne units in Korea are Detachment 39 and 112th Signal attached to SOCKOR, neither of which have a 74D position. A more viable option is to request 1st Bn, 1st SFG in Okinawa. You can catch a hop to Korea, and frequent training and TDY there as well.

Suicidal if you ask me. With your MOS you can go pretty much every where so try to get an assignment to Korea. Perhaps call to your branch manager is in place!

Is her PhD going to be useful in the US? Many foreign educations do not transfer well and if she stays there for this and then needs to stay there for work, which is important to you, an Army career or living with her in Korea? You. may need to make a difficult decision in the near future or at the very least have this discussion with her.