  We introduce a simple model for implementing the concepts of quasi-energy and
parametric resonances (PRs) in systems with the $\mathcal{PT}$ symmetry, i.e.,
a pair of coupled and mutually balanced gain and loss elements. The parametric
(ac) forcing is applied through periodic modulation of the coefficient
accounting for the coupling of the two degrees of freedom. The system may be
realized in optics as a dual-core waveguide with the gain and loss applied to
different cores, and the thickness of the gap between them subject to a
periodic modulation. The onset and development of the parametric instability
for a small forcing amplitude ($V_{1}$) is studied in an analytical form. The
full dynamical chart of the system is generated by systematic simulations. At
sufficiently large values of the forcing frequency, $\omega $, tongues of the
parametric instability originate, with the increase of $V_{1}$, as predicted by
the analysis. However, the tongues following further increase of $V_{1}$
feature a pattern drastically different from that in usual
(non-$\mathcal{PT}$)\ parametrically driven systems: instead of bending down to
larger values of the dc coupling constant, $V_{0}$, they maintain a direction
parallel to the $V_{1}$ axis. The system of the parallel tongues gets dense
with the decrease of $\omega $, merging into a complex small-scale structure of
alternating regions of stability and instability. The cases of $\omega
\rightarrow 0$ and $\omega \rightarrow \infty $ are studied analytically by
means of the adiabatic and averaging approximation, respectively. The cubic
nonlinearity, if added to the system, alters the picture, destabilizing many
originally robust dynamical regimes, and stabilizing some which were unstable.
