For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach.