Transverse wall oscillations have shown to suppress kinetic energy and skin-friction drag in turbulent channel flows. The performance of this flow control strategy in providing optimal drag reduction has been shown to crucially depend on the frequency and amplitude of oscillations. We analyze the robust performance of this control strategy in the presence of parametric uncertainties in the amplitude and phase of oscillations that can be caused by imperfect implementation. Such parametric uncertainties enter the linearized dynamics in a multiplicative manner and can dramatically affect the mean-square properties of flow fluctuations. We adopt an input-output approach to show that certain levels of parametric uncertainty can indeed violate stability conditions. For those that preserve stability, we observe a series of adverse effects that range from the promotion of turbulence to the increase in drag and changes to the optimal drag-reducing frequency of wall oscillations.