= Solution
Write $m=|m|e^{i\theta}$. <Parity> reverses the spatial coordinates and must exchange the two chiral <Weyl spinors>. One convenient choice is
$$
\boxed{
P:\quad
\psi_L(t,\mathbf x)\mapsto e^{-i\theta}\psi_R(t,-\mathbf x),
\qquad
\psi_R(t,\mathbf x)\mapsto e^{i\theta}\psi_L(t,-\mathbf x)}.
$$
The identities $\sigma^0=\bar\sigma^0$ and $\sigma^i=-\bar\sigma^i$ exchange the two kinetic terms after $\mathbf x\mapsto-\mathbf x$. The mass bilinear $m\bar\psi_R\psi_L$ maps to its <complex conjugate> $m^*\bar\psi_L\psi_R$, so the two mass terms are exchanged and the action is invariant. More generally, independent unit phases $a,b$ may multiply the two transformations provided $b^*a=m^*/m$.
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