I think I'm missing something quite basic here but consider the process:
$$ e^- + e^+ \rightarrow 2\gamma$$
Fermions have opposite parity to antifermions so the parity quantum number before the process is $P=-1 \times (-1)^L$ where $L$ is the relative orbital angular momentum, which should vanish in the zero momentum frame where the collision is head on. So we start with $P=-1$. The photons on the other hand have the same parity as each other so have parity $P=+1$ (after again noting that they must be heading in opposite directions to each other in the ZMF so have $L=0$).
I think my reasoning about the angular momentum must be wrong, but I don't see why. (Additionally, the same argument says that if this is $L:0\rightarrow 0$ then charge congugation $C$ is not a conserved quantity either.)