Claiming your parent as your dependent does not legally obligate you to pay their medical bills, although if you paid them you can enter them into the medical expense section on your own tax return.
When you claim an adult dependent you get a $500 credit for other dependents. You do not receive a stimulus check for claiming them---and the dependent adult does not receive a stimulus check because dependents are not eligible for stimulus checks. So you need to be sure of what you are doing when you say that a parent is your dependent and know that you are really providing over half their support.
CREDIT FOR OTHER DEPENDENTS
No, claiming your parent on your tax return means that you 'provided over half of their support'.
It does not make you responsible for their medical bills (by the IRS), although if you contributed to their medical bills, this is a form of 'support'.
Click this link for more info on What Does Financially Support Mean?
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Thank you for the responses. The forms she received from the hospital say to state whether she can be counted as a dependent by someone else. I wonder what the purpose of asking that is. The amount in tax savings is not worth the hassle of being hounded by medical bill collectors.