 Parity transformation of a Dirac field

With the mostly-minus Minkowski metric, $|{\eta }_P|=1$ and the usual Dirac adjoint, this transformation makes $\bar{\psi }\psi$ even and $\bar{\psi }{\gamma }^5\psi$ odd under parity. The latter sign follows from anticommutation of ${\gamma }^0$ and ${\gamma }^5$ and does not depend on whether the convention includes an imaginary unit in ${\gamma }^5$.