Collective excitations, such as vortex-antivortex and dark solitons, are among the most fascinating effects of macroscopic quantum states. However, two-dimensional (2D) dark solitons are unstable and collapse into vortices due to snake instabilities. Making use of the optical bistability in exciton-polariton microcavities, we demonstrate that a pair of dark solitons can be formed in the wake of an obstacle in a polariton flow resonantly supported by a homogeneous laser beam. Unlike the purely dissipative case where the solitons are gray and spatially separate, here the two solitons are fully dark, rapidly align at a specific separation distance, and propagate parallel as long as the flow is in the bistable regime. Remarkably, the use of this regime allows us to relax the phase fixing constraints imposed by the resonant pumping and to circumvent the polariton decay. Our work opens very wide possibilities for studying new classes of phase-density defects which can form in driven-dissipative quantum fluids of light.