  We study parametrically driven quantum oscillators and show that, even for
weak coupling between the oscillators, they can exhibit various many-body
states with broken time-translation symmetry. In the quantum-coherent regime,
the symmetry breaking occurs via a nonequilibrium quantum phase transition. For
dissipative oscillators, the main effect of the weak coupling is to make the
switching rate of an oscillator between its period-2 states dependent on the
states of other oscillators. This allows mapping the oscillators onto a system
of coupled spins. Away from the bifurcation parameter values where the period-2
states emerge, the stationary state corresponds to having a microscopic current
in the spin system, in the presence of disorder. In the vicinity of the
bifurcation point or for identical oscillators, the stationary state can be
mapped on that of the Ising model with an effective temperature $\propto
\hbar$, for low temperature. Closer to the bifurcation point the coupling can
not be considered weak and the system maps onto coupled overdamped Brownian
particles performing quantum diffusion in a potential landscape.
