Illegal downloads on wireless router?

I live on my parents' property, and I accidentally downloaded something illegal off of their internet service through the wireless router. They now have a large bill. I'm hoping they can dispute it, thinking that some stranger did it. I just want to see if it would be possible for it to be traced back to my computer by the internet company? There are three households on one router, and each have a different "name". Would the company be able to trace it back to mine, titled "LISA"? I'm willing to pay the bill if I can find a way, I just don't want my family to find out it was my fault. Right now, they are thinking it was someone else. If it helps, we have Charter.