If they are your parents, say "in my parent's name".
Note: If it's only in the name of your father or only in the name of your mother, than "in my parent's name" (one parent) is correct.
But if it's in both of their names, then it's "in my parents' names."
I rarely would say "my parent" to mean "my mother" or "my father." I would say "a car in my dad's/father's name" or "a car in my mom's/mother's name."
Did you mean one parent or both?