  We study superconductivity in spin-orbit-coupled systems in the vicinity of
inversion symmetry breaking. We find that due to the presence of spin-orbit
coupling, fluctuations of the incipient parity-breaking order generate an
attractive pairing interaction in an odd-parity pairing channel, which competes
with the s-wave pairing. We show that applying a Zeeman field suppresses the
s-wave pairing and promotes the odd-parity superconducting state. Our work
provides a new mechanism for odd-parity pairing and opens a route to novel
topological superconductivity.
