  We investigate tests of the parity P and the combined parity and
charge-conjugate CP symmetries from differential angular distributions of
$J/\psi$ decaying into the lowest-lying baryon pairs at BESIII and the
next-generation super tau-charm facilities (STCFs). Large corrections from $Z$
and $W$ exchange induced parity violating effects are found for $J/\psi$ decays
with large logarithms resummed up to $\mathcal{O}(\alpha_s)$. The
parity-violating asymmetries on the production and the decay sides of $J/\psi$
are both found to be of $\mathcal{O}(10^{-4})$, thus barely observable with the
10 billion $J/\psi$ events currently collected at BESIII. Nevertheless, these
asymmetries utilizing the current BESIII data already permit a determination of
the weak mixing angle with an absolute uncertainty $\delta s_w^2\approx0.08$,
corresponding to the first determination of $s_w^2$ at the $J/\psi$ threshold.
While limited by statistics currently, STCFs are estimated to improve this
precision by a factor of $\sim\,20$ to $\delta s_w^2\approx0.004$ within one
year based on luminosity rescaling. We also obtain the 95% confidence level
upper bounds on the electric dipole moments of the octet baryons, which are of
$\mathcal{O}(10^{-18})\,e{\rm\,cm}$ for BESIII and
$\mathcal{O}(10^{-19})\,e{\rm\,cm}$ for STCFs. These bounds are improved by two
to three orders of magnitude in comparison with the only existing one on
$\Lambda$ from Fermilab. The method discussed in this work also paves a way for
a first and direct measurement of the $\Xi$ and $\Sigma$ electric dipole
moments.
