$p$ is a homogeneous polynomial. Show that $p^{-1}(a)$ is a submanifold of $\mathbb{R}^n$ if $a\neq 0$. The submanifolds of $a>0$ are diffeomorphic.

The thing about being homogeneous of degree $m$ is that $p(tx) = t^mp(x)$ for $t >0$. Using this should give you an idea for an explicit diffeomorphism from $p^{-1}(t^ma)$ to $p^{-1}(a)$.

It is straightforward to see that $p^{-1}(a)$ and $p^{-1}(-a)$ are diffeomorphic, so it suffices to show that for $0<a<b$, the level sets $p^{-1}(a)$ and $p^{-1}(b)$ are diffeomorphic, but as you noticed $0$ in the only possible critical value of $p$. In particular, there is no critical value of $p$ between $a$ and $b$ and the statement follow from the standard critical non-crossing theorem in Morse theory.