  One class of solutions to the strong CP problem relies on generalized parity
symmetries. A minimal model of this type, constructed by Babu and Mohapatra and
based on a softly broken parity symmetry, has the remarkable property that
effective QCD vacuum angle $\bar\theta$ vanishes up to one-loop order. We
compute the leading two-loop contributions to $\bar\theta$ in this model and
estimate subleading contributions. In contrast to previous estimates, we argue
that $\bar \theta$ is not suppressed by the weak scale, and we find
contributions of order $10^{-3}$-$10^{-2}$ multiplying unknown mixing angles
and phases. Thus the model does not generically address the strong CP problem,
but it might be made consistent with $\bar\theta<10^{-10}$ in some corners of
parameter space. For such non-generic parameters, $\bar\theta$ is still likely
to be just below present bounds, and therefore provides the dominant source of
hadronic EDMs. We discuss the resulting EDM phenomenology.
