This, for me, is the critical point here.
Longshanks, you clearly have a better grasp of finances than your parents, well done to you. You can’t tell your parents how to spend their money but they also can’t expect to tell you how to spend yours.
If I was you, I’d say that I am happy to pay rent for the time you live with them, and it’s up to them what they do with this money, but this money should not tie you in to some loan they are choosing to take out. In a year or two, you might want to move out, and the last thing you want is some expectation from your parents that you will continue to pay ‘your share’ of their debt.
To reassure you, legally you won’t be responsible for any of their debt, as presumably your name won’t be involved with the bank, but you don’t want to fall out with your parents over it or feel obligated, so make it clear up front now. In writing, preferably.
(BTW, to the Americans in this thread, in the UK at age 16 you can leave school, leave home, join the army, have sex, rent a home (if you have an adult guarantor), choose your own doctor, consent to medical treatment, drink beer with a meal in a restaurant or pub, and get married. At 17 you can buy a firearm and ammunition, yes, you read that right!)