After an accident, your car may end up in a storage yard or at a tow facility. The question then becomes who pays for these costs, including tow costs.
If you have collision coverage, your insurance company should be paying the tow and storage facility. They will ask you to “release the vehicle” to them. You should comply with this request as quickly as possible. No one gains anything, other than the tow yard, if the car stays in storage. Your insurance company will then pay for the towing and storage.
If you do not have collision coverage, and you are at fault for the accident, you get to pay it. Get your car out of there as quickly as possible. The charges will add up quickly.
If you do not have collision coverage and the other person is at fault, then their insurance company SHOULD pay for it. I say SHOULD because they do not always pay for it. They should pay for the tow charge and the storage for a reasonable time. If they do not know who you are, and you do not know who they are, then you should move the vehicle. Once they know about you and your claim, it becomes, in my opinion, their responsibility. If they are going to stop paying for storage, they should let you know in writing and at least 5 days before they stop paying.