We show that a parallel differential form $\Psi$ of even degree on a Riemannian manifold allows to define a natural differential both on $\Omega$($M$) and $\Omega$*($M$,$TM$), defined via the Frolicher-Nijenhuis bracket. For instance, on a Kaehler 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 Spin(7)-manifold, respectively. We calculate the cohomology groups of $\Omega$*($M$) and give a partial description of the cohomology of $\Omega$*($M$,$TM$). This is joint work with Hong Van Le and Lorenz Schwachhoefer.