  We study the dissipative dynamics of a single quantum harmonic oscillator
subjected to a parametric driving with in an effective Hamiltonian approach.
Using Liouville von Neumann approach, we show that the time evolution of a
parametrically driven dissipative quantum oscillator has a strong connection
with the classical damped Mathieu equation. Based on the numerical analysis of
the Monodromy matrix, we demonstrate that the dynamical instability generated
by the parametric driving are reduced by the effect of dissipation. Further, we
obtain a closed relationship between the localization of the Wigner function
and the stability of the damped Mathieu equation.
