Parameter sensitivities of prices for derivative contracts play an important role in model calibration as well as in quantification of model risk. In this paper a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented. Variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model differentiated with respect to the model parameters are employed. Superconvergent approximations to second and higher derivatives of prices with respect to the price process' state variables are extracted from approximate, computed prices with low, C⁰ regularity by postprocessing. The extracted numerical sensitivities are proved to converge with optimal rates as the mesh width tends to zero. Numerical experiments for uni- and multivariate models with sparse tensor product discretization confirm the theoretical results.