Equimultiplicity of topologically equisingular families of parametrized surfaces in mathbb C³

We provide a positive answer to Zariski's conjecture for families of singular surfaces in $\mathbb C^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization of the Whitney equisingularity of one parameter families of $\mathcal A$ finitely determined map-germs $f_t: (\mathbb C^2,0) \to (\mathbb C^3,0),$ in terms of the constancy of only one invariant, the Milnor number of the double point locus.