Summary: In this paper, we show that a parallel differential form \(\Psi\) of even degree on a Riemannian manifold allows to define a natural differential both on \(\Omega^\ast(M)\) and \(\Omega^\ast(M, T M)\), defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel \(4\)-form on a \(G_2\)- and \(\mathrm{Spin}(7)\)-manifold, respectively. We calculate the cohomology groups of \(\Omega^\ast(M)\) and give a partial description of the cohomology of \(\Omega^\ast(M, T M)\).