At the death of a parent with $8000 of credit card debt, will the children be responsible?

No. Send copy of death certificate. If there is money in the estate they may file a claim against it.

There is a caveat here - if you are a signer on the account, they may try to get you to pay the debt, especially if you ever signed your name on any credit card slips. If you are a joint credit card holder, the credit card companies can come after you for the debt. Determine which is the case and go forward accordingly. If the credit card companies write off the debt, they may issue a 1099-C, cancellation of debt, and if the estate is not closed, you will need to file paperwork with the IRS that it is not income to your deceased mother and no taxes are due.