We consider the large pump-detuning limit of the optical parametric oscillator equations which scale as either a focusing or defocusing parametrically forced nonlinear Schrödinger equation. Linearizing about the pulse or front solutions leads to a non-sectoral, non-self-adjoint linear eigenvalue problem which we analyse via a non-perturbative method, localizing the point spectrum to regions of the complex plane. We establish H ¹ linear decay estimates for the semi-group and employ renormalization group methods to demonstrate the nonlinear asymptotic stability of the solutions to small H ¹ perturbations of initial conditions.