No, children are not responsible for the debts of their parents. However, if deceased parents left behind any property whatsoever, their estate would be responsible for paying their debts. How much money goes to the creditors versus the family would be a matter of state law.
As long as the children were not separately liable (e.g., by co-signing), generally only the parents' estates are liable for their pre-mortem debts. To enforce those obligations, the creditors must file and serve their claims to the personal representative, administrator or notice agent prior to the second anniversary of the parent's date of death or, if earlier, the otherwise applicable statute of limitations.
You can accelerate this cut off to a four month period by publishing notice and providing specific written notice to "ascertainable" creditors. In some cases, however, it is worth waiting out the two years. Notice that there are specific requirements to notify the DOR (state taxes) and DSHS (generally if Medicaid paid for longterm care).
There is one exception to my knowledge under state law: the situation where the estate is insolvent (canot pay all creditors fully after dealing with secured creditors and paying (1) administrative expenses, (2) funeral and last illness expenses and (3) federal taxes). In that unhappy circumstance, in probate or if the non-probate creditor claims procedure is utilized, the personal representative may be required to recover additional funds from beneficiaries of certain non-probate assets. (Even though an IRA would be exempt for most purposes, the IRS can theoretically collect for back income taxes from it.)
I am sorry to hear of your loss and this has to just make it worse. That said, credit card debts are not secured. (unless it is a HELOC, possibly). Not secured means that if an older person ran up their credit cards and did not pay them, then the creditor is not likely to get paid. That is a risk they take. If you are being contacted by a collector telling you that you have to pay this, particularly Philip Cohen and Associates, when you write back to them, be sure to send a copy to the Attorney General, Consumer Protection Division. Collectors like this love to give you legal advice that strongly suggests that they should be paid first.
If you don't have a probate attorney working on this, it would likely be worth it to at least gather the information about your parent's estate (assets and liabilities) and get a consult.