The present thesis provides a survey of results and techniques from the Langlands program which allow for the computation of the \(\ell\)-adic cohomology of local systems on Shimura varieties in terms of automorphic representations. We implement this strategy for the group \(G=\mathrm{GSp}_4\) and obtain results on the \(\ell\)-adic cohomology of local systems on Siegel modular threefolds. We then specialize these results to the case of square-free parahoric level in order to obtain explicit computations using results of R\"osner on depth zero parahoric restriction. We further specialize these results to obtain explicit computations of the \(\ell\)-adic cohomology of local systems on the moduli of principally polarized Abelian surfaces with full level 2 structure, and resolve conjectures of Bergstr\"om-Faber-van der Geer in this case.