If you're paying out of pocket to have the parts on your vehicle replaced then yes you can request that the auto repair shop return the old replaced parts to you after the repair is done.
However if the auto repairs are being done under warranty then you cannot keep the replaced parts from the auto repair.
If I take my car into an auto repair shop to have parts replaced I do request the old parts be returned to me so I can be sure that the parts were actually replaced with new ones.
I've had some shops say they repaired and replaced some parts when in fact they didn't replace those parts or even touch them and I was still charged for it.
So that's why I insist every shop I pay out of pocket too for the repairs to return my old parts to me.
An honest auto shop will gladly return the parts to you.