The electric and magnetic field components $E_i$ and $B_i$, defined from the electromagnetic field tensor $F_{\mu\nu}$ are $$E_i=cF_{0i},~~B_i=-\frac{1}{2}\epsilon_{ijk}F^{jk}.$$ Since $\epsilon_{ijk}$ is an invariant pseudotensor, it transforms as $\epsilon_{ijk}\xrightarrow{{\rm parity}}-\epsilon_{ijk}$. It can also be quickly checked that $F^{jk}\xrightarrow{{\rm parity}} F^{jk}$.
But this leads to a wrong conclusion that $B_i\xrightarrow{{\rm parity}}-B_i$. This means $\vec{B}$ is not an axial vector which is definitely false. What is the mistake?