  Photonic systems with time-varying modulations have attracted considerable
attention as they allow for the design of non-reciprocal devices without the
need for an external magnetic bias. Unlike time-invariant systems, such
modulations couple modes with different frequencies. Here, we discuss how this
coupling and particle-hole symmetry may lead to the resonant interaction of
positive and negative frequency oscillators. To illustrate this idea, we
analyze a dispersive spacetime crystal described by a Drude-Lorentz model with
a travelingwave modulation. Our findings demonstrate that the interaction
between positive and negative frequency bands can induce parametric
instabilities under certain conditions, stemming from the interplay between
dispersion and spacetime modulations. In particular, we find that material
dispersion creates the conditions for the formation of instabilities for
arbitrarily small modulations speeds in the absence of dissipative channels.
