We derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems. This analysis extends the results in [T. Blass and L. A. Romero, SIAM J. Control Optim., 51 (2013), pp. 1099-1127], where, under the assumption that the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this sum to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this approach leads to an enormous simplification. We give two examples of systems with rank one forcing, including the problem of stochastically forced Faraday waves.