  Experiments searching for the electric dipole moment (EDM) of the electron
$d_e$ utilise atomic/molecular states with one or more uncompensated electron
spins, and these paramagnetic systems have recently achieved remarkable
sensitivity to $d_e$. If the source of $CP$ violation resides entirely in the
hadronic sector, the two-photon exchange processes between electrons and the
nucleus induce $CP$-odd semileptonic interactions, parametrised by the Wilson
coefficient $C_{SP}$, and provide the dominant source of EDMs in paramagnetic
systems instead of $d_e$. We evaluate the $C_{SP}$ coefficients induced by the
leading hadronic sources of $CP$ violation, namely nucleon EDMs and $CP$-odd
pion-nucleon couplings, by calculating the nucleon-number-enhanced $CP$-odd
nuclear scalar polarisability, employing chiral perturbation theory at the
nucleon level and the Fermi-gas model for the nucleus. This allows us to
translate the ACME EDM limits from paramagnetic ThO into novel independent
constraints on the QCD theta term $|\bar \theta| < 3 \times 10^{-8}$, proton
EDM $|d_p| < 2 \times 10^{-23}\,e\,{\rm cm}$, isoscalar $CP$-odd pion-nucleon
coupling $|\bar g^{(1)}_{\pi NN}| < 4 \times 10^{-10}$, and colour EDMs of
quarks $|\tilde d_u - \tilde d_d| < 2 \times 10^{-24}\,{\rm cm}$. We note that
further experimental progress with EDM experiments in paramagnetic systems may
allow them to rival the sensitivity of EDM experiments with neutrons and
diamagnetic atoms to these quantities.
