We consider the dynamics associated with topological solitons (localized structures) of the optical parametric oscillator which models the parametric exchange of energy between optical fields at a fundamental and second harmonic frequency. Simulations show that this nonlinear interaction can support stable front structures as well as localized, bistable solitary wave solutions. We perform a systematic study of the bifurcation structure and stability analysis of both solitary wave and front solutions which arise. The stability analysis is carried out for the onset of instability which arises from a Ginzburg-Landau description as well as a modified Swift-Hohenberg description at resonance. The analysis, which is carried out in 1-D, can be utilized in predicting the dynamical behavior in 2-D systems. Further, the theoretical conclusions provide important practical predictions which are verified via extensive numerical simulations.