If your parents aren't incapacitated, the simplest solution is to have them sign checks and pay their bills from the hospital. (You may find that they use an automated bill-paying service or have their bills paid automatically online. In that case, you won't need to worry about debts piling up.)
If your parents aren't able to help you with this, your options depend on whether you have a durable power of attorney (DPOA) in place for each of them.
Ideally, before the accident occurred, you and your parents saw an attorney together and prepared this legal document, which gives you the ability to access accounts, pay bills, and make financial, legal, and real estate decisions on their behalf.
If you already have a DPOA, contact your parents' banks and creditors. Let them know that your parents have been incapacitated and that you're acting as their designated decision-maker. You should have no trouble paying your parents' bills from their bank accounts.
If you don't have a DPOA and your parents are still mentally competent, ask them to sign the forms immediately. Someone on the hospital staff should be able to help you get the right forms and arrange for a notary to visit your parents' rooms to get the papers signed and notarized.
If your parents aren't mentally competent, your options without this legal document are fairly limited. One option, if you have the means and can collect the mail from your parents' house, is to go ahead and pay their bills yourself -- it's not illegal to pay someone else's bills.
However, if you want to pay the bills from their accounts, you're probably out of luck. Because of privacy laws, unless you have a power of attorney, their banks won't give you any information or access to the accounts.
Finally, if you don't have durable power of attorney forms signed and are concerned that your parents will be incapacitated for a long period of time, you can go to court to petition to be named as their guardian or conservator. Unfortunately, this can be a long and expensive process, so be prepared.