  We study a cavity-QED setup consisting of a two-level system coupled to a
single cavity mode with two-photon relaxation. The system dynamics is modeled
via a Lindblad master equation consisting of the Rabi Hamiltonian and a
two-photon dissipator. We show that an even-photon relaxation preserves the
$Z_2$ symmetry of the Rabi model, and provide a framework to study the
corresponding non-Hermitian dynamics in the number-parity basis. We discuss the
role of different terms in the two-photon dissipator and show how one can
extend existing results for the closed Rabi spectrum to the open case.
Furthermore, we characterize the role of the $Z_2$ symmetry in the
excitation-relaxation dynamics of the system as a function of light-matter
coupling. Importantly, we observe that initial states with even-odd parity
manifest qualitatively distinct transient and steady state behaviors, contrary
to the Hermitian dynamics that is only sensitive to whether the initial state
is parity-invariant. Moreover, the parity-sensitive dynamical behavior is not a
creature of ultrastrong coupling and is present even at weak coupling values.
