If you're married, but still live with your parents and they are currently paying for your college education and anything else you might need, can they claim you as dependent on their taxes? Also, if your spouse does not claim you on their taxes is this possible?

Yes this is possible. Provided you did not make in excess of the amount you would have to file income taxes for yourself.

Couple of other points:

1. Assuming you are a full time student then you must be either under age 24 or have income less than $3,650 in order for your parents to claim you as a dependent.

2. If your parents claim you as a dependent then your wife must file her own tax return as married filing separately (she may not file jointly).