Recent experimental advances have made it possible to detect individual quantum jumps in open quantum systems, such as the tunneling of single electrons in nanoscale conductors or the emission of photons from nonclassical light sources. Here, we investigate theoretically the statistics of photons emitted from a microwave cavity that is driven resonantly by an external field. We focus on the differences between a parametric and a coherent drive, which either squeezes or displaces the cavity field. We employ a Lindblad master equation dressed with counting fields to obtain the generating function of the photon emission statistics using a theoretical framework based on Gaussian states. We then compare the distribution of photon waiting times for the two drives as well as the g(2) functions of the outgoing light, and we identify important differences between these observables. In the long-time limit, we analyze the factorial cumulants of the photon emission statistics and the large-deviation statistics of the emission currents, which are markedly different for the two drives. Our theoretical framework can readily be extended to more complicated systems, for instance, with several coupled microwave cavities, and our predictions may be tested in future experiments.