  We obtain estimates on the first-order Malliavin derivative of mild
solutions, evaluated at fixed points in time and space, to a class of parabolic
dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular,
such equations are driven by multiplicative Wiener noise and the nonlinear
drift term is the superposition operator associated to a locally Lipschitz
continuous function satisfying suitable polynomial growth bounds. The main
arguments rely on the well-posedness theory in the mild sense for stochastic
evolution equations in Banach spaces, monotonicity, and a comparison principle.
