And it appears like in the US you can get one for free, to monitor your credit.
So I would request your free credit report. See what it says. Maybe satellite bills never show up on there, in which case this isn't something to worry about for now. That being said, I would still get it out of your name, especially if you're planning on moving. Put that on your list of things to do.
Different people have different views on bill paying. That's normal and natural. Some carry credit card balances. Some would never do that in a million years. So it's OK that you and your parents differ on how to manage this. The key is that things in your name are managed the way you want. If it was in their name they could manage it any way THEY wanted. So if they want to pay late, then it should be in their name. No emotion, no confusion, just simple.
Money is the #1 cause of arguments in relationships. It becomes emotional. We all have different views on what is "right". So the best thing is to organize the finances so it's perfectly clear. If they're paying a bill, their name should be on that bill.