  We prove an invariance principle for the two-dimensional lattice parabolic
Anderson model with small potential. As applications we deduce a Donsker type
convergence result for a discrete random polymer measure, as well as a
universality result for the spectrum of discrete random Schr\"odinger operators
on large boxes with small potentials. Our proof is based on paracontrolled
distributions and some basic results for multiple stochastic integrals of
discrete martingales.
