  Parity violation in the gravitational sector is a prediction of many theories
beyond general relativity. In the propagation of gravitational waves, parity
violation manifests by inducing amplitude and/or velocity birefringence between
right- and left-circularly polarized modes. We study how the stochastic
gravitational wave background can be used to place constraints on these
birefringent effects. We consider two model scenarios, one in which we allow
birefringent corrections to become arbitrarily large, and a second in which we
impose stringent theory priors. In the former, we place constraints on a
generic birefringent gravitational-wave signal due to the current non-detection
of a stochastic background from compact binary events. We find a joint
constraint on birefringent parameters, $\kappa_D$ and $\kappa_z$, of
$\mathcal{O}(10^{-1})$. In the latter scenario, we forecast constraints on
parity violating theories resulting from observations of the future upgraded
LIGO-Virgo-KAGRA network as well as proposed third-generation detectors. We
find that third-generation detectors will be able to improve the constraints by
at least two orders of magnitude, yielding new stringent bounds on parity
violating theories. This work introduces a novel and powerful probe of
gravitational parity violation with gravitational-wave data.
