I wouldn't say it's wrong, it's just different. If you write "parent's" then you're talking about one parent. "That parent's child is misbehaving a lot at school". You're only refering to a particular parent.
It follows that "parents'" would be plural. "The parents' child is misbehaving at school". Now you'd be referring to both parents.
The question-setter is making the reasonable assumption that we do not talk about my parent's car referring to a car belonging to only one parent.
If the car belongs to one parent then it is either my father's car or my mother's car.
While technically correct, my parent's car is very unlikely to be used naturally - and that is what the question-setter is expecting you to understand.