If one has quit one's job to move in to care for one's parents and has no income, can that person be claimed as their parents dependent?

It's possible, but I think you would be better off working and claiming yourself or you will have zip zero nothing for yourself when it comes time to collect your own SS payments. Set up a care contract with a lawyer so your parents can pay you something.

The only way your parent can claim you as a Dependent is if they are paying more than half of your care. Check with the IRS or a Tax Accountant for more information.