  The isovector time-reversal- and parity-violating pion-nucleon coupling
$\bar{g}_{\pi}^{(1)}$ is uniquely sensitive to dimension six interactions
between right-handed light quarks and the Standard Model Higgs doublet that
naturally arises in left-right symmetric models. Recent work has used the
$\bar{g}_{\pi}^{(1)}$-induced one-loop contribution to the neutron electric
dipole moment $d_n$, together with the present experimental $d_n$ bound, to
constrain the CP-violating parameters of the left-right symmetric model. We
show that this and related analyses are based on an earlier meson theory $d_n$
computation that is not consistent with the power-counting appropriate for an
effective field theory. We repeat the one-loop calculation using heavy baryon
chiral perturbation theory and find that the resulting $d_n$ sensitivity to
$\bar{g}_{\pi}^{(1)}$ is suppressed, implying more relaxed constraints on the
parameter space of the left-right symmetric model. Assuming no cancellations
between this loop contribution and other contributions, such as the leading
order EDM low-energy constant, the present limit on $d_n$ implies
$|\bar{g}_{\pi}^{(1)}| \lesssim 1.1\times 10^{-10}$.
