= Solution
The <chiral projectors> $P_{L,R}=(1\mp\gamma^5)/2$ obey $\gamma^\mu P_L=P_R\gamma^\mu$, while the <Dirac adjoints> are $\bar\psi_L=\bar\psi P_R$ and $\bar\psi_R=\bar\psi P_L$. Consequently the cross terms vanish and the <vector current> is
$$
\bar\psi\gamma^\mu\psi=\bar\psi_L\gamma^\mu\psi_L+\bar\psi_R\gamma^\mu\psi_R.
$$
The vector interaction preserves <fermion chirality> at a vertex and couples equally to left and right components. This statement about the vertex does not assert that a <fermion> <mass> term preserves <chirality>. Under <parity>, left and right components are exchanged; their equal coefficients allow a parity-invariant contraction with a true vector field.
An axial vector must couple to the <axial current> to obtain the analogous parity-invariant interaction. Since $\gamma^5\psi_L=-\psi_L$ and $\gamma^5\psi_R=\psi_R$,
$$
\bar\psi\gamma^\mu\gamma^5\psi=-\bar\psi_L\gamma^\mu\psi_L+\bar\psi_R\gamma^\mu\psi_R,
\qquad
\boxed{\mathcal L_A=-g_A(-\bar\psi_L\gamma^\mu\psi_L+\bar\psi_R\gamma^\mu\psi_R)A_\mu.}
$$
The relative sign changes; the overall sign is conventional. A pure axial interaction can itself conserve <parity> when $A_\mu$ transforms as an axial vector. Simply contracting the original <vector current> with an axial-vector field would instead produce a parity-odd interaction.
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