Can power of attorney be held responsible if principal causes car accident?

The fact that you hold someone's power of attorney does not make you legally responsible for their actions. You are legally responsible for your own actions, of course. If as part of a settlement for the car accident you were supposed to issue a payment from an account of theirs that you have power of attorney over and you don't, you could both be in trouble over that, but the car accident itself would not be your fault.