  We theoretically study NQR and NMR spectra in the presence of odd-parity
multipoles originating from staggered antiferromagnetic and antiferroquadrupole
orderings. For the $f$-electron metal, CeCoSi, which is a candidate hosting
odd-parity multipoles, we derive an effective hyperfine field acting on Co
nucleus generated from electronic origin multipole moments at Ce ion under zero
and nonzero magnetic fields. We elucidate that emergent odd-parity multipoles
give rise to sublattice-dependent spectral splittings in NQR and NMR through
the effective hyperfine coupling in the absence of the global inversion
symmetry. We mainly examine behaviors of the NQR and NMR spectra in three
odd-parity multipole ordered states: a $y$-type magnetic toroidal dipole order
with a staggered $x$-type antiferromagnetic structure, an $xy$-type electric
toroidal quadrupole order with a staggered $x^2-y^2$-type antiferroquadrupole
structure, and a $z$-type electric dipole order with a staggered
$3z^2-r^2$-type antiferroquadrupole structure. We show that different
odd-parity multipole orders lead to different field-dependent spectral
splittings in NMR, while only the $xy$-type electric toroidal quadrupole order
exhibits the NQR spectral splitting. We also present possible
sublattice-dependent spectral splittings for all the odd-parity multipole
orders potentially activated in low-energy crystal-field levels, which will be
useful to identify odd-parity order parameters in CeCoSi by NQR and NMR
measurements.
