In the Fermi weak theory we have the fermion bilinears which look like 

$$
V_\mu = \bar{\psi} \gamma_\mu\psi
$$
$$
A_\mu = \bar{\psi} \gamma_\mu \gamma_5 \psi
$$

Under a parity transformation 

$$
x = (x_0, \vec{x}) \rightarrow \tilde{x} = ( x_0, - \vec{x})
$$

The fields transform like

$$
V^\mu(x) \rightarrow V_\mu(\tilde{x})
$$
$$
A^\mu(x) \rightarrow - A_\mu(\tilde{x})
$$

Why do the contravariant indices transform to covariant indices as well as a coordinate transformation? I thought it would have something to do with the fact that you also have to transform the actual vector components under a coordinate/parity transformation, but I don´t know how to formalize it starting from the explicit form of the bilinears.