In this paper we prove that the density [math] of the solution of a white-noise-driven parabolic stochastic partial differential equation (SPDE) satisfying a strong ellipticity condition is [math] Lipschitz continuous with respect to (w.r.t.) [math] and [math] Lipschitz continuous w.r.t. [math] for all [math] . In addition, we show that it belongs to the Besov space [math] w.r.t. [math] . The proof is based on the Malliavin calculus of variations and on some refined estimates for the Green kernel associated with the SPDE.