The Lorentz invariant term $\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu}F_{\sigma\rho}$ is not parity invariant. To show this one needs to find the parity transformation property of $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. Under parity i.e., $x^0\to x^0, x^i\to -x^i$, $$\partial_0\to\partial_0,~~\partial_i\to -\partial_i~~\text{and}~~ A_0\to A_0,~~A_i\to -A_i.\tag{1}$$

$\bullet$ To find the transformation property of $F_{\mu\nu}$, do I need to check each component of $F_{\mu\nu}$ separately? That would seem rather inelegant. I also do not want to change it into $\textbf{E}$ and $\textbf{B}$ and use their parity transformation properties.

$\bullet$ How do I show that $\epsilon^{\mu\nu\sigma\rho}$ change sign under parity? I have no clue why should it even change sign under parity.

$\bullet$ What about the parity property of gluon field strength $G_{\mu\nu}^a=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu+g f^{abc}A_\mu^b A_\nu^c$? To find the relations in Eq.(1), I used the behaviour of $\textbf{E}$ and $\textbf{B}$ from classical electrodynamics. Where do we get the behaviour of gluon fields $A_\mu^\nu$ under parity?