In this study we have theoretically investigated the effect of parallel superposition of modulation on the stability of single-layer Newtonian and viscoelastic flows down an inclined plane. Specifically, a spectrally based numerical technique in conjunction with Floquet theory has been used to investigate the linear stability of this class of flows. Based on these analyses we have demonstrated that parallel superposition of modulation can be used to stabilize or destabilize flow of Newtonian and viscoelastic fluids down an inclined plane. In general at low Reynolds number Re (i.e. O(1)) and in the limit of long and O(1) waves the effect of dynamic modulation on the stability of viscoelastic flows is much more pronounced; however, relatively large modulation amplitudes are required to achieve significant stabilization/destabilization. In addition, the dependence of the most dominant modulation frequencies on Re and the Weissenberg number We have been identified. Specifically, it has been shown that for Newtonian flows low-frequency modulations are destabilizing and the most dominant modulation frequency scales with 1/Re. On the other hand, for viscoelastic flows in the absence of fluid inertia low-frequency modulations are stabilizing and the most dominant modulation frequency scales with 1/We. In finite-Re viscoelastic flows the most dominant destabilizing modulation frequency scales with 1/Re while the most stabilizing modulation frequency scales with 1/WeRe. Finally, it has been demonstrated that the mechanism of both purely elastic and inertial instabilities in flows down an inclined plane is unchanged in the presence of dynamic modulation.