  Theory of the superconducting parity transition is extended by incorporating
the vortex degree of freedom. We employ the bilayer Rashba model representing
locally noncentrosymmetric layered superconductors and derive the
Ginzburg-Landau free energy functional. This formulation reveals the parity
transition, where the even-parity superconducting state changes to the
odd-parity one upon increasing the magnetic field under the vortex states. The
H-T phase diagram of CeRh${}_{2}$As${}_{2}$ is quantitatively reproduced and a
novel superconducting state with a meron (half-skyrmion) lattice pseudospin
texture is predicted.
