I'd properly write it as, "neither of my parents are teachers". (A) makes perfect grammatical sense. (B) follows the logic of your question, but the grammar is wrong. Even if you wrote, "my father and mother are not teachers" it makes simple grammatical sense but is still not entirely right. When using negatives there's a convention to avoid inclusive words like "and". So my rewrite is "neither my father nor my mother are teachers". Hope this helps, though I didn't actually answer the question. ;)

B! both means "the two" or "the one as well as the other"