My elderly parents are coming to visit me in the U.S. They purchased some travel insurance policies in their country. However, I am not familiar with the terms and deductibles of those policies. I would like to know the financial ramifications of them possibly needing medical assistance that their policies will not cover. E.g., if I had to take them to the ER and they needed a surgery, could the hospital go after me for the costs?
The short version is that you are not responsible for your relatives' debts. If your parents come to the US and incur massive medical bills, the hospital/insurance company can try to claim the back from the parents, but not you. And if your parents have no assets in the United States, it will generally be difficult to impossible to enforce payment of those debts overseas.
Travel insurance is generally sold as a standalone product, where the insurance company takes on all the risk and does not require any sort of co-signed or guarantor. Since your parents apparently arranged their own insurance, there's no way you could have accrued any legal liability for it.
One potential exception: If your parents should happen to die in the United States, there is a possibility that you would be held responsible for their medical bills under filial responsibility laws, but these vary from state to state and are generally not enforced unless there's evidence of ill will (eg. you siphoning your parents' bank accounts dry).
Due to the ridiculous cost of medical treatment in the US, travel insurance to the US is usually expensive, doubly so if you are elderly. So in your shoes, I would be more concerned with checking that the coverage is adequate. Otherwise there is a low but non-zero risk of your parents being refused admission to a hospital, especially if it's not directly life-threatening medical emergency.