"How do you write your dad's name?"
Translation:¿Cómo se escribe el nombre de tu papá?
No. The question is asking how the name of your father is written (spelled). So it takes on the form of "se escribe _ nombre". Just like if I wanted to ask you, "How do you spell/write the word happy in English?" it would be "¿Cómo se escribe la palabra 'happy' en inglés?"
"escribes" was accepted but I wish I had come up with "se escribe." I think it's a better answer.