The teacher is right: $1{\color{red}/}0$ is undefined. (if he said 'doesn't have an answer', then he is being somewhat sloppy)
However, the father is right: $1{\color{green}/}0 = \infty$.
But also, the title is right: $1{\color{blue}/}0 = \mathrm{NaN}$.
The problem is that the topic fits into what I believe to be a significant gap in mathematics education: people aren't taught syntax and mathematical grammar at all, so they don't have the ability to make precise statements about what they mean -- or even to know that it's an issue!
(I've added color to emphasize that I mean three different things in those three statements!)
The first version of division is what is taught in elementary school; the teacher is right on that point. $1{\color{red}/}0$ is a syntax error: $(1,0)$ isn't in the domain of ${\color{red}/}$, and so it is illegal to write the expression evaluating ${\color{red}/}$ at $(1,0)$.
The second version, however, is the division of projective numbers. The projective numbers are very useful for algebraic purposes, and even for some analytic purposes: e.g. it can be convenient to have $\tan$ be projective-valued, so that $\tan(\pi/2) = \infty$. I was being a little forgiving when I said the father was right, though -- I find it more likely he was thinking about the extended real numbers (but not knowing that by name!), and simply made a common mistake.
The third version is back to ordinary division, but in a syntax/semantics based on something like partial functions or composition of relations. A rough description is that in so far as functions $\{ * \} \to \mathbb{R}$ correspond to elements of $\mathbb{R}$, the partial function $\{ * \} \to \mathbb{R}$ with empty domain corresponds to $\mathrm{NaN}$.
On this last point, note that to some extent we force students to actually think in terms of this family of concepts with notation like $1 \pm \sqrt{2}$ and $x^3/3 + C$, and questions like "What is the domain of $1/(1-x)$?". But IMO, these notions are somewhat incongruous with what students are actually taught about functions.