If your parent dies with a mountain of credit-card debt, that's not your problem. It's your parent's estate, not you, that has to settle his debts. The worst that might happen is that after the estate pays off the creditors, you have nothing left to inherit. There are certain circumstances, however, in which you might be liable for parental debt.
A person's debts are paid off as the estate goes through probate. The executor notifies known creditors by mail and also typically must publish a notice in the newspapers to alert any other creditors. That sets the clock ticking: creditors who fail to file by your state's probate deadline -- typically four to six months -- don't get paid. They're also out of luck if the estate has no money to pay them. They may ask the children or other beneficiaries of the estate to pay, but they can't force you to cough up the cash.
If you and your parent had a joint credit card account, you're legally liable for anything on the card his estate can't pay off. Even if none of the charges are yours, you're still on the hook. You're also liable if you made him an authorized user on your own card.
Even when their name isn't on a credit card, children can be held liable if they ripped off their parents. For example, if someone took advantage of managing his parent's finances to put a few indulgences on her card, or applied for extra cards in her name, all that debt is the child's responsibility. The card company will have to prove it was the child's spending, but if they establish that, he's stuck.
After a parent dies, his creditors may tell you that you have a moral obligation to pay his debts. If you're guilty or grief-stricken enough to do so, that's a happy day for the creditors. Some scam artists will try to exploit your grief by claiming debts that don't exist. The safest move is to tell them to stop calling you and direct the calls to your parent's executor. It's often wise not to give them any information about yourself, your credit cards or anything else that they can use against you.