My parents put my sister and I on their bank accounts; are we liable for any nursing home costs?

My dad’s retirement is too much to qualify for Medicaid assistance for nursing home care; but not enough to pay for them both (neither in good health; will both qualify for Veterans Home, but that cost is beyond their retirement by $3k a month). Will the Veterans Home be able to force my sister and I to pay the balance because my parents put our names on their accounts? I don’t want to have to give up my own retirement to pay for this. Of course we would sell their assets for their care—but can they come after my own assets if they are not in my parents’ names?