= Solution
Use metric signature $+---$ and the <parity> matrix $P^\mu{}_{\nu}=\operatorname{diag}(1,-1,-1,-1)$. If a <Dirac field> transforms as $\psi'(x)=\eta_P S_P\psi(x_P)$, the chain rule changes the spatial derivatives' signs. Covariance of its <Dirac equation> therefore requires
$$
S_P^{-1}\gamma^0S_P=\gamma^0,\qquad S_P^{-1}\gamma^iS_P=-\gamma^i.
$$
The <Clifford algebra> gives precisely these identities for $S_P=\gamma^0$. In an irreducible Dirac representation any other solution differs by a scalar phase, because its ratio with $\gamma^0$ commutes with all <gamma matrices>. With $|\eta_P|=1$, the Dirac <mass> bilinear and <kinetic term> consequently obey
$$
\bar\psi'(x)\psi'(x)=\bar\psi(x_P)\psi(x_P),\qquad
\bar\psi'(x)i\gamma^\mu\partial_\mu\psi'(x)
=\bar\psi(x_P)i\gamma^\mu\partial_\mu\psi(x_P).
$$
\b[The factor $\gamma^0$ compensates the sign reversal of spatial derivatives.] A transformation containing only the argument change would not preserve the massive <Dirac equation>. This establishes <parity> symmetry of the free or vector-coupled Dirac theory; it does not make a chiral <weak interaction> <parity> symmetric.
For <charge conjugation>, transpose the adjoint <Dirac equation> to obtain $(i\gamma^{\mu T}\partial_\mu+m)\bar\psi^T=0$. Multiplying by $C$ gives the original <Dirac equation> for $\psi^c=C\bar\psi^T$ if
$$
\boxed{C\gamma^{\mu T}C^{-1}=-\gamma^\mu,\qquad
C^{-1}\gamma^\mu C=-\gamma^{\mu T}.}
$$
One may choose $C$ unitary; in four dimensions a conventional choice in a standard Dirac or chiral basis is $C=i\gamma^2\gamma^0$, with $C^T=-C$. Its overall phase and $\eta_C$ do not affect the following bilinear transformation. Reordering the Grassmann <fermion> fields gives, for a color matrix $X$ commuting with the <gamma matrices>,
$$
\overline{\psi^c}\gamma^\mu X\psi^c=-\bar\psi\gamma^\mu X^T\psi.
$$
Thus a vector color current changes sign and transposes its color generator under <charge conjugation>.
Write the Hermitian color connection as $\mathcal A_\mu=A_\mu^aT^a$. The interaction coming from $D_\mu=\partial_\mu+ig\mathcal A_\mu$ is $-g\bar\psi\gamma^\mu\mathcal A_\mu\psi$. Its charge-conjugation symmetry requires $\mathcal A_\mu^C=-\mathcal A_\mu^T$. <Parity> acts on the Lorentz index as on a vector connection, so combining the two gives
$$
\boxed{\mathcal A_\mu^{CP}(x)=-P_\mu{}^\nu\mathcal A_\nu^T(x_P).}
$$
In components, $\mathcal A_0\mapsto-\mathcal A_0^T(x_P)$ and $\mathcal A_i\mapsto+\mathcal A_i^T(x_P)$. The transpose is a color-space transpose, independent of the spinor matrix $C$. Intrinsic <fermion> phases cancel between a field and its adjoint.
Define the matrix <gauge field strength> unambiguously by
$$
\mathcal F_{\mu\nu}=\partial_\mu\mathcal A_\nu-\partial_\nu\mathcal A_\mu+ig[\mathcal A_\mu,\mathcal A_\nu].
$$
The derivative terms transform with the expected <parity> factors and a minus transpose. For the commutator, the crucial identity is $[X^T,Y^T]=-[X,Y]^T$, so the nonlinear term has the same transformation as the derivative terms. This is the <CP transformation of non-Abelian field strength>:
$$
\boxed{\mathcal F_{\mu\nu}^{CP}(x)=-P_\mu{}^\alpha P_\nu{}^\beta\mathcal F_{\alpha\beta}^T(x_P).}
$$
For example, $\mathcal F_{0i}\mapsto+\mathcal F_{0i}^T(x_P)$ while $\mathcal F_{ij}\mapsto-\mathcal F_{ij}^T(x_P)$. With $\operatorname{tr}(T^aT^b)=\frac12\delta^{ab}$, the color sum in the theta operator is twice a <trace> of two field strengths. The two charge-conjugation minus signs cancel, and transposition reverses the <trace> factors without changing their <trace>. The four <parity> matrices contracted with the <Levi-Civita symbol> contribute $\det P=-1$. Hence
$$
\boxed{\left[\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^aF_{\rho\sigma}^a\right]^{CP}(x)
=-\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^aF_{\rho\sigma}^a(x_P).}
$$
\b[The theta operator is <CP> odd; a generic fixed nonzero coefficient violates <CP>.] Setting the coefficient to zero removes this term. Although the density is locally a total derivative, nontrivial gauge topology means that it need not be irrelevant to the quantum theory. If a properly normalized theta angle is identified periodically, invariance at special values equivalent to their negatives is a separate global question; it does not change the <CP> odd transformation of the local operator.
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