  Linear mechanical systems with time-modulated parameters can harbor
oscillations with amplitudes that grow or decay exponentially with time due to
the phenomenon of parametric resonance. While the resonance properties of
individual oscillators are well understood, those of systems of coupled
oscillators remain challenging to characterize. Here, we determine the
parametric resonance conditions for time-modulated mechanical systems by
exploiting the internal symmetries arising from the real-valued and symplectic
nature of classical mechanics. We also determine how these conditions are
further constrained when the system exhibits external symmetries. In
particular, we analyze systems with space-time symmetry where the system
remains invariant after a combination of discrete translation in both space and
time. For such systems, we identify a combined space-time translation operator
that provides more information about the dynamics of the system than the
Floquet operator does, and use it to derive conditions for one-way
amplification of traveling waves. Our exact theoretical framework based on
symmetries enables the design of exotic responses such as nonreciprocal
transport and one-way amplification in dynamic mechanical metamaterials, and is
generalizable to all physical systems that obey space-time symmetry.
