No. In almost every case you are not responsible for your parents' debt when they die, and you do not inherit it. If a debt collector tells you that you have to pay your dead mother's credit card or your father's medical bill out of your own money, they are usually wrong, and sometimes breaking the law.[1]
Debt belongs to the person who borrowed the money. When your parent dies, their debt becomes a claim against their estate, which is everything they owned. The estate pays what it can. What it can't pay usually goes unpaid. There are a few real situations where you can be liable, and this page names each one plainly so you can tell whether it applies to you. Most of the time, it doesn't.[3]