Photonic crystals have revolutionized the field of optics with their unique dispersion and energy band gap engineering capabilities, such as the demonstration of extreme group and phase velocities, topologically protected photonic edge states, and control of spontaneous emission of photons. Time-variant media have also shown distinct functionalities, including nonreciprocal propagation, frequency conversion, and amplification of light. However, spatiotemporal modulation has mostly been studied as a simple harmonic wave function. Here, we analyze time-variant and spatially discrete photonic crystal structures, referred to as spatiotemporal crystals. The design of spatiotemporal crystals allows engineering of the momentum band gap within which parametric amplification can occur. As a potential platform for the construction of a parametric oscillator, a finite-sized spatiotemporal crystal is proposed and analyzed. Parametric oscillation is initiated by the energy and momentum conversion of an incident wave and the subsequent amplification by parametric gain within the momentum band gap. The oscillation process dominates over frequency mixing interactions above a transition threshold determined by the balance between gain and loss. Furthermore, the asymmetric formation of momentum band gaps can be realized by spatial phase control of the temporal modulation, which leads to directional radiation of oscillations at distinct frequencies. The proposed structure would enable simultaneous engineering of energy and momentum band gaps and provide a guideline for implementation of advanced dispersion-engineered parametric oscillators.