The insurance company that will be paying will based on what type of damages are claimed? Even though the other party is not "at fault" for this accident, it is possible that person's insurance company may have to pay out for this claim. However, your question leads me to believe you want to know if you will be responsible for this accident. As the driver, you will be responsible. However, insurance coverage follows the vehicle first. So, assuming the insurance company of the owner of the vehicle finds it acceptable for you to be driving that person's vehicle, their coverage will be "primary." Any insurance coverage you have will be "excess" or "secondary."