Is an adult child responsible for debt on checking acct of parent if both names are on the acct at time of parent death?

No, not necessarily. It is not the same as a husband and wife being 'joint' on an account. The holder of the debt would have to prove that you (the adult child) benefitted in some way from whatever it was that was purchased with the parent's check -or- that you were a co-signer to some contract or sales agreement for which the parent was paying. If any money is allegedly due and owing to a claimant, they will have to file a lien against your parent's estate and prove to the court that it is a legitimate debt.