The point is that, locally, any Du Val singularity can be realized as a double cover of a smooth surface. This means that there exist analytic coordinates such that the germ of singularity has the form $$x^2=f(y, \, z),$$ that is the embedding dimension is $3$. In other words, even if the surface is not globally embeddable in $\mathbb{A}^3$, an analytic neighbourhood of the singularity always is.
A good reference is [Barth-Peters-Van De Ven, Compact Complex Surfaces, Springer 1984], see in particular Lemma 3.8 of Chapter III.