  The weak interaction does not conserve parity, which is apparent in many
nuclear and atomic phenomena. However, thus far, parity nonconservation has not
been observed in molecules. Here we consider nuclear-spin-dependent parity
nonconserving contributions to the molecular Hamiltonian. These contributions
give rise to a parity nonconserving indirect nuclear spin-spin coupling which
can be distinguished from parity conserving interactions in molecules of
appropriate symmetry, including diatomic molecules. We estimate the magnitude
of the coupling, taking into account relativistic corrections. Finally, we
propose and simulate an experiment to detect the parity nonconserving coupling
using liquid- or gas-state zero-field nuclear magnetic resonance of
electrically oriented molecules and show that $^{1}$H$^{19}$F should give
signals within the detection limits of current atomic vapor-cell magnetometers.
