  Background: A recent coupled reaction channel (CRC) study shows that the
enhanced oscillation of the elastic $^{16}$O+$^{12}$C cross section at backward
angles is due mainly to the elastic $\alpha$ transfer or the core exchange.
Such a process gives rise to a parity-dependent term in the total elastic
$S$-matrix, an indication of the parity dependence of the $^{16}$O+$^{12}$C
optical potential (OP).
  Purpose: To explicitly determine the core exchange potential (CEP) induced by
the symmetric exchange of the two $^{12}$C cores in the elastic
$^{16}$O+$^{12}$C scattering at $E_{\rm lab}= 132$ and 300 MeV, and explore its
parity dependence.
  Method: $S$-matrix generated by CRC description of the elastic
$^{16}$O+$^{12}$C scattering is used as the input for the inversion calculation
to obtain the effective local OP that contains both the Wigner and Majorana
terms.
  Results: The high-precision inversion results show a strong contribution by
the complex Majorana term in the total OP of the $^{16}$O+$^{12}$C system, and
thus provide for the first time a direct estimation of the parity-dependent
CEP.
  Conclusions: The elastic $\alpha$ transfer or exchange of the two $^{12}$C
cores in the $^{16}$O+$^{12}$C system gives rise to a complex parity dependence
of the total OP. This should be a general feature of the OP for the light
heavy-ion systems that contain two identical cores.
