A fast and parallel Stroud-based stochastic collocation method for statistically characterizing electromagnetic interference and compatibility (EMI/EMC) phenomena on loaded multiscale platforms with uncertain system configurations and subject to variable electromagnetic excitations is described. The proposed method uses a previously developed hybrid time domain integral equation based field-cable-circuit to carry out deterministic EMI/EMC simulations permitting the statistical characterization of pertinent observables. The number of simulations required by the proposed method is far fewer than those needed by Monte-Carlo methods. The proposed method is used to characterize cable-induced coupling onto PC cards located in shielding enclosures. Both the hybrid simulator and the stochastic collocation code execute with near-full efficiency on distributed memory clusters.