Can a parent be liable if their child is a ward of the state and they were claimed on their income tax in Kentucky?

Yup...if they are a ward of the state they didn't qualify as a dependent, and doing so subjects you to, of course, return of any amounts you benefited ont he return, penality for doing so and underpayment, interest on the monies and even criminal charges for fraud. Of course, the unsaid but maybe even worse thing is, your then an untrusty taxpayer and nothing concerning your account (and likely prior years too), will happen without substantial review.