Is the party who caused a car accident entitled to request the old damaged part after paying for a new one, or is the amount paid a compensation for the damage, and the car owner has the choice to repair or restore it?

What appears evident is that the one who hit you and paid the value of the new rear bumper has the right to take the old bumper, because it has become his property. You do not have the right to combine the old bumper with the price of the new one, as this is a combination of the substitute and the thing replaced.