morally you but legally them, if you're not their legal sponsor.

may be but not sure, because not all the time immigration officer investigate these cases,
"if they don't pay it will go in collection, in their next visit they might be denied at port of entry."

"001" (sorry in advance, but some info you are providing are incomplete or not accurate)

do not know morally who is responsible, but legally, the person who signed the document is responsible.

Usually when you go to doctor or ER, there is one form that states that party responsible for payments if no insurance or insurance didn't cover. Same logic is applied to minors where parents would be responsible.

My suggestion - do not sign any documents and ask your parents to sign it - both during the entry and exit after visit.
All they will do is send you bills or call you at the address and phone no provided buy you.
Once it goes to collection - they will cal you and ask about your parents. Tell them politely, that they have left the country and you can provide them with their internal no and request them not to call you. The call will stop after few tries.

I do not think it will have any impact on future visits. Without SSN it is very hard to track this.
These days, medical bills in your credit report are not even considered major.

BTW, sometimes the hospital might send you a letter requesting that you write an application so that all or major part of bills be waived. Do that too if there is an option.

"frustu" - do not worry. Hopefully it will never happened, but in case it happened, no hospital can deny services to any individual brought into ER because of no insurance or immigrant status. But I SUGGEST you to have that coverage although it doesn't cover much.

Also, (advice from my doctor friend who works at one of the major gov hospital in big city), if you want the ER see patient quick, complain about chest pain including all other issues. They will admit them in minutes and paperwork can be done later. Not suggesting you do this, just pointing out for those who might need one day.

Do you I have to submit Affidavit of Support while inviting your parents?

The answer is "YOU"...now there are several loopholes, which many people here will let you know

The best you can do is buy travel insurance for some fix amount. It’s really worth it trust me. But in case you don’t want to buy it for whatever reasons and you have to take someone who does not have SSN to the hospital, while filling out the form provide all incorrect info. Without proper ID of the patient no one can validate this anyways.