Ok. There are two separate issues here. First, your parents financial state and secondly the debt they have run up in your name.
Let's deal with the debt first. You have two options. Either report your parents for identity theft and fraudulently taking out the credit in your name, or accept the debt and pay it. If you chose to report them it will be *your* job to prove it wasn't you who took out the credit. The penalties for this sort of thing can be very severe including huge fines and jail time. If you chose to pay the debt off then speak with the debt collection agency. They will set up a payment plan with you so you can pay it back slowly. Thank hard about which of these options sounds better to you, because if you can't get your parents to pay for it (which i think they should but they don't seem to be in a position to) then they are your only choices.
Secondly is your parents situation. Are you willing to help them get their lives in order? If so, there are things you can do. First, speak with their various creditors, explain the situation and work out payment plans that will be more affordable. Then write out a budget for them. Regarding the electricity, (i got my start in the gas & electric business for one of the UK "Big six" suppliers in their debt collection & warrants departments) call and ask what help they offer for vulnerable customers. There's fuel direct which means the cost of the bill will be taken directly from your parents benefits (which i presume they get as they don't work) so they will always have fuel. If they currently have a prepay meter ask for it to be taken back out and fuel direct applied for due to risk of "self disconnection". There may even be specific teams who can offer additional help/grants/etc.
Go to the CAB and ensure your parents are getting all of the benefits they are entitled to and also to get advice regarding the debts if you need to.
Ultimately though, your parents must be willing to change, if they refuse to sort things out, or make noises about wanting to but don't change then you can't make them. thank long and hard about your relationship with them and make absolutely sure they don't have access to any of your documents etc so they can't do this to you again.