To answer your original question...NO...using your parents credit cards will not affect your credit rating, unless you are a co-signer on the card. A co-signer is different than an "authorized user". A co-signer can be held accountable financially and legally for the debt incurred on a card, an authorized user can not. More than likely, you are just an authorized user and your parents are co-signers for each other. Also, using your parent's credit card does NOT make you a dependent (no more than you using MY credit card would make you MY dependent). The only thing that makes you "dependent" is your age, and whether your parents claim you on their tax returns (sometimes parents do this even if they DON'T provide more than half of the child's income...but that's another issue..)
As far as Ohio's residency issues, you should just call the DMV (or whatever office handles that stuff up there) and ASK THEM what they mean by "independent from parents". It may just mean you live away from home...or they may go by the IRS standards. You'll have to make your decisions on what to do then.
Hope this helps