You don't have to go all into sentence structure and everything. Just say that "lives" refers to a singular subject while "live" would refer to a plural subject.
The boy and girl, who lives with her parents, yadda yadda
vs.
The boy and girl who live with her parents.
Then there's what you were probably going to go into - the fact that "who lives with her parents" is an interjection in the sentence, shown by the commas.
Either way, from the way I read it they didn't live together.