We derive an analytical solution for the angular caustic in parabolic mirrors, extending geometric optics beyond the paraxial approximation. By solving the ray-mapping equations for the parabolic profile, we determine the topology of the critical ray manifold. We find that the focal singularity is not a static point but a continuous angular distribution extending toward grazing incidence in the near-field regime. This singularity is classified as a stable $A_2$ fold catastrophe, distinct from the degenerate focus of the linear limit, providing analytical bounds for the finite macroscopic window of the two-ray interference regime. Furthermore, we evaluate the geometric stability of the cavity to establish bounds for experimental setups. By defining an intermediate parameter window governed by the short-wavelength limit ($\lambda /b\ll 1$) and a near-field observation distance ($0.30\le a/b\le 0.40$), we identify the operational regime where the non-paraxial caustic shift macroscopically clears the diffraction limit while remaining robust against mechanical error amplification. These results provide a geometric framework for determining macroscopic spatial structures and optimal trap positioning for cavity QED, optical trapping, and semiclassical field theory within strongly curved mirror geometries.