Student or Learner
"I drive a car that is in parent's name or in the parent's name." Which one is correct?
"Dad's car" (if speaking to a brother or sister) or "my dad's car."
The Prime Minister.
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?
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.