You can say that if the conversation has already established that you're talking about occupations.
So your father is a dentist.
No, it's my mom who is a dentist.
Oh, I see. What is your dad then?
Even so, I don't recommend it. It sounds too much like you are defining the entire person by his occupation.
A response to "What is your father" could be anything from "A terrific fellow" to "About 1.9 meters" to "A doctor," depending on the context of the conversation.
[a business writer, not a teacher]
Student or Learner