"What are your parents' names?"
Here's why:
* "Parents'" is the possessive form of "parents," indicating that the names belong to them.
* "Are" is the correct verb form for plural nouns like "names."
You could also say:
* "What are the names of your parents?" This is a slightly more formal way of phrasing the question.
It's important to remember that using proper grammar makes your communication clear and understandable.