We discuss a scheme for parametric amplification of the quantum fluctuations of the electromagnetic vacuum in a three-dimensional microwave resonator and report preliminary measurements to test its feasibility. In the present experimental scheme, the fundamental mode of a microwave cavity is non-adiabatically perturbed by modulating the index of refraction of a nonlinear optical crystal enclosed therein. Intense, multi-GHz laser pulses as those delivered by a mode-locked laser source impinge on the crystal to accomplish the n-index modulation. We theoretically analyze the process of parametric generation, which is related to the third-order nonlinear coefficient χ (3) of the nonlinear crystal, and assess suitable experimental conditions for generating real photons from the vacuum. Second-order nonlinear processes are first analyzed as possible source of spurious photons in quantum vacuum experiments when an ideal, mode-locked laser source is considered. The combination of a crystal non-null χ (2) coefficient and a real mode-locked laser system, i.e. featuring offset-from-carrier noise and unwanted secondary oscillations, is also experimentally investigated in the second part of the work, paving the way for future experiments in three-dimensional cavities.