Are adult children responsible for their deceased parent's unpaid nursing home bills?

No. Not unless they agreed in writing to be responsible for those bills. A parent's estate is responsible for paying their debts whether those debts are for a nursing home, credit cards or utility bills. Their property cannot be distributed until the debts are paid. If there are not enough assets in the estate the creditors are out of luck. The exception is mortgages. If the mortgage isn't paid the bank will take possession of the property by foreclosure.