When your car is towed to a tow yard, typically the insurance investigating or covering the property damage claim (i.e. your car), will pay for towing cost and resonable storage fees. If the insurance company has declared your vehicle a total loss, they will usually need your permission to pick it up at the tow yard. However if no insurance company has declared your vehicle a total loss, you need to mitigate (limit the amount of storage cost that accrue) your damages. The insurance company is not required to pay for any thing, and if you just leave the vehicle you will be stuck paying the tow yard owner storage fees from your property damage settlement. This could be substantial.