It is useful to understand the effect of a parity inversion on a Dirac spinor.
Again work with the Dirac equation and its parity inverted form in which
(the same for the vector potential).
commutes with itself but anticommutes with the
, it works fine.
(We could multiply it by a phase factor if we want, but there is no point to it.)
Therefore, under a parity inversion operation
the third and fourth components of the spinor change sign while the first two don't.
Since we could have chosen
, all we know is that
components 3 and 4 have the opposite parity of components 1 and 2.