After moving elders to a nursing home, how does one pay for the bills left behind at home?

Is your name on that personal loan?

No I don't sign for anything

Just guessing here, but to tell you the truth, as someone recently said to me "it is only numbers spit out by a computer anymore" as I attempted to deal with bills and no one would help. Even medical billing is not done by the hospitals or doc offices, but by billing companies as far flung as across the country or indeed across the world. So to tell the truth, unpaid they may ruin their credit. Does anyone CARE about that at this point? In four years they will likely mostly be forgiven after the bill collectors get tired of you all.
Not to say someone cannot take them to court and win a case when they don't show up for the court date, and they can then put a lein on their home, so that any sale of it will pay whatever judgements are rendered, but comes a point when it just is impossible to deal with the bills of another, that you do not owe. WHATEVER YOU DO, do not pay them YOURSELF out of your funds. Do what you can as POA for your parents, but their immediate needs and care first. Once in care anyone who placed a lein on the home will be in line with medicaid recovery, so assets of the home will be gone.
You cannot get blood out of a turnip as they used to say. Do not make any payments yourself; this kicks the four year point for most states back to day one. Do not speak to people. Tell them your parents are "in care" when they are, and unable to speak. As to now, tell them your parents are very elderly and unable to speak with them on the phone, and bills will be paid in order sent as they are able to afford them, which is finally not much. At some point, hang up.
If you have POA to pay there bills this is all more complicated. You may at some point need the advice of a lawyer. Spend their money for THAT, not for the medical bills.

If you have not co-signed on anything, you do nothing, especially if it is medical or CC bills. If they are not on Medicaid, I would recommend that they apply!

Yes all the equity was taken out and then property values have fallen. It is an unsecured personal loan through a company. They still have very good credit. They used to have a very nice lifestyle. They have just outlived all the savings.

You need to get them on Medicaid, even if its just for health insurance at this point. What Medicare doesn't pay Medicaid does. If you feel they are ready, then try Medicaid for a NH. The house will not count as an asset. Any savings. Ira, bonds, insurance policies with cash in value, etc will have to be liquidated for their care. Prepaid funerals can be set up as part of the spenddown. There SS will go towards their care. The VA income may stop because both that and Medicaid is government money.

Right before my inlaws went into 2 separate care facilities, they were not only broke but upside down on their home mortgage and owed tens of thousands on cc's. We were scrambling to figure out what to do as my StepFIL's entire SS check paid only the interest on the second mortgage. We couldn't communicate with the mortgage company because the loan was in my stepFIL's name and he gave no one PoA but he had Parkinsons, Lewy Body and cognitive decline. Their son/my husband the PoA and other siblings agreed to just pay a small amount toward the monthly mortgage payment so we could fend off foreclosure, which was inevitable.

We liquidated everything in their home, cancelled every service and paid minimums on their medical bills from their own funds until we could get permanent placement for them. We didn't "stiff" anyone except the cc companies which was an overwhelming amount and the debt now owned by collection companies.

Your LOs will need to apply for Medicaid. They seem to have nothing (not even owning their home apparently) so they've got nothing to lose and their care will be the same as those who pay out of pocket.

Regarding debts, if a company takes you to court, then they can not request money for the debt to be taken out of their social security checks. It's against the law. I don't know if this helps you, but it's helpful to know, just in case.