We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplicative ergodic theorem with an exponential rate of convergence. The assumptions are satisfied for a large class of parabolic PDEs, including the 2D Navier–Stokes and complex Ginzburg–Landau equations perturbed by a non-degenerate bounded random kick force. As a consequence of this ergodic theorem, we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system. In particular, we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible. The proof relies on a refined version of the uniform Feller property combined with some contraction and bootstrap arguments.