arXiv:2504.04788v2 Announce Type: replace Abstract: We theoretically investigate the dynamics of parametrically driven soliton crystals (PDSC) and their associated frequency combs in doubly resonant cavities with quadratic and cubic nonlinearities. We show that, in a regime with strong pump--signal walk-off where the homogeneous state is unstable, noise-seeded dynamics relaxes toward stable, equally spaced multi-soliton states. At fixed detuning, the driving strength acts as the primary control parameter for the soliton number. Pump signal walkoff extends pump depletion from a local perturbation to a global constraint, enabling long-range soliton interactions; in combination with the pump phase, this mechanism stabilizes the crystal's equal spacing and preserves comb coherence. Additionally, we identify a novel nonlinear state in parametric soliton crystals in which circulating solitons periodically alternate their intensities and group velocities, a phenomenon we term the {\it soliton-pursuing} state. Furthermore, due to the phase-selective nature of the optical parametric process, we show how different configurations of soliton phases determine the optical frequency combs, enabling odd-harmonic and subharmonic-like combs.