  A microscopic calculation and symmetry argument reveal superconductivity in
the vicinity of parity-violating magnetic order. An augmented cluster magnetic
multipole order in a crystal lacking local space inversion parity may break
global inversion symmetry, and then, it is classified into an odd-parity
multipole order. We investigate unconventional superconductivity induced by an
odd-parity magnetic multipole fluctuation in a two-dimensional two-sublattice
Hubbard model motivated by Sr$_2$IrO$_4$. We find that even-parity
superconductivity is more significantly suppressed by a spin-orbit coupling
than that in a globally noncentrosymmetric system. Consequently, two odd-parity
superconducting states are stabilized by magnetic multipole fluctuations in a
large spin-orbit coupling region. Both of them are identified as $Z_2$
topological superconducting states. The obtained gap function of
inter-sublattice pairing shows a gapped/nodal structure protected by
nonsymmorphic symmetry. Our finding implies a new family of odd-parity
topological superconductors. Candidate materials are discussed.
