Reeder and Yu have constructed in a uniform way certain supercuspidal representations of p-adic groups called "epipelagic representations", using invariant theory studied by Vinberg et al. In the function field case, we will realize these epipelagic representations as local components of automorphic representations, and construct the corresponding Langlands parameters, i.e., local systems over P^1 minus two points. These local systems can be computed explicitly for classical groups, and they give new families of local systems (with monodromy in all types of groups, classical or exceptional) that are expected to be rigid.