The effect of a finite-bandwith pump on the parametric instability is considered. First, the temporal problem with unequal damping rates is considered, with bandwidth modelled by a Kubo-Anderson process. Exact results are obtained for both the amplitudes and power in each decay mode, with an interesting asymmetry found. Next, the spatial problem with unequal group velocities is investigated for the cases of 1) finite homogeneous interaction region, and 2) linear density profile. Thresholds and growthrates are found including both the inhomogeneity and bandwidth effects. When values typical of laser fusion parameters are inserted, it is found that bandwidth may be an important method of stabilizing Raman and Brillouin scattering.