Suppose your efforts work, and the car begins to move forward out of the mud. As it does so, the force of the car on the rope is

equal to the force of the rope on the car.

Explanation:

When dealing with ropes the force that it exerts and the force being exerted on it is always equal as long as the rope doesn't snap/break. Therefore, the force of the car on the rope is equal to the force of the rope on the car. The fact that the car is moving due to your efforts simply means that you are exerting more force than all the forces that are pushing in the opposite direction, mainly gravity and friction. Since the force you are exerting is greater it allows you to move the car, but as long as the rope remains intact then the force on the rope is equal on both sides.