Here are some ways to respond, depending on the context and your comfort level:
Formal:
* Mis padres se llaman [nombre del padre] y [nombre de la madre]. (My parents' names are [father's name] and [mother's name].)
* El nombre de mi padre es [nombre del padre] y el de mi madre es [nombre de la madre]. (My father's name is [father's name] and my mother's name is [mother's name].)
Informal:
* Mi papá se llama [nombre del padre] y mi mamá se llama [nombre de la madre]. (My dad's name is [father's name] and my mom's name is [mother's name].)
* Se llaman [nombre del padre] y [nombre de la madre]. (They're called [father's name] and [mother's name].)
If you don't want to share their names, you could say:
* Lo siento, prefiero no decirlo. (Sorry, I prefer not to say.)
* No me acuerdo. (I don't remember.)
* No es algo que me guste compartir. (It's not something I like to share.)
Remember, the most appropriate response will depend on your relationship with the person asking the question.