  We pinpoint the conditions necessary for discrete time crystal (DTC)
formation in fully connected spin-cavity systems from the perspective of
parametric resonance by mapping these systems onto oscillator like models. We
elucidate the role of nonlinearity and dissipation by mapping the periodically
driven open Dicke model onto effective linear and nonlinear oscillator models,
while we analyze the effect of global symmetry breaking using the
Lipkin-Meshkov-Glick model with tunable anisotropy. We show that the system's
nonlinearity restrains the dynamics from becoming unbounded when driven
resonantly. On the other hand, dissipation keeps the oscillation amplitude of
the period-doubling instability fixed, which is a key feature of DTCs. The
presence of global symmetry breaking in the absence of driving is found to be
crucial in the parametric resonant activation of period-doubling response. We
provide analytic predictions for the resonant frequencies and amplitudes
leading to DTC formation for both systems using their respective oscillator
models.
