You may not claim the mileage rate for using a borrowed car. You may only claim your actual out-of-pocket expenses, like gas & tolls & parking. If you paid your parents to use their car, you may claim the car "rent" as a business expense. However to be perfectly legit, you should written them a check in 2016.
Technically, they are required to report the "personal property rental income" on their tax return. But they are allowed to deduct up to 54 cents per mile as offsetting expenses.
