  It is known that the nuclear-spin-dependent parity-violating contributions to
the NMR shielding and the nuclear spin-rotation tensors (${\bf \sigma}^{PV}$
and ${\bf M}^{PV}$, respectively) are formally related each other within the
non-relativistic (NR) regime. Such a formal relationship is not any longer
valid within the relativistic domain. A new more general formal relationship,
that is valid within the relativistic framework is shown here, being developed
through the use of the LRESC model. The formalism of polarization propagators
is applied to write the different contributions to both properties within both
regimes, relativistic and NR. In the relativistic regime the Dirac-Coulomb
Hamiltonian was selected as the unperturbed Hamiltonian. Theoretical
developments together with results of calculations performed on the H$_2X_2$
series of molecules ($X =$ $^{17}$O, $^{33}$S, $^{77}$Se, $^{125}$Te and
$^{209}$Po) show that also within the relativistic regime there is a close
relationship between the parity-violation contributions to both properties. In
particular, spin-dependent contributions are the most important in the
four-component calculations of electroweak effects on the isotropic values of
both tensors, ${\bf \sigma}^{PV}$ and ${\bf M}^{PV}$, being also responsible
for the breakdown of the previously mentioned NR relationship among them. This
last relationship is still fulfilled when the scalar-relativistic effects are
considered.
