We investigate the contributions to $\mu - e $ conversion in nuclei in a supersymmetric model with left-right symmetry, motivated by the new data on neutrino oscillations. We study the dependence of the conversion rate on the various parameters of the model, and show that light-mass or large $\tan \beta$ scenarios are severely restricted. We analyse the effect of several popular mecahnisms of neutrino mixing on the conversion rate as well as the influence of the right-handed scale on the conversion rate. We compare the conversion rate to the branching ratio for $\mu \rightarrow e \gamma$ and discuss their relative accessibility at future experiments, their sensitivity to various parameters of the model, as well as their relative importance in providing signals for new Physics.