The intermediate quarkonium state $\chi_2(^3P_2)$ in electron-positron annihilation to proton and antiproton as well as in antiproton-proton annihilation to electron and positron can produce backward-forward asymmetry, when populated through two photon exchange. We use the dispersion relation method, which permits to express the asymmetry in terms of partial widths of quarkonium decay. The asymmetry dependence on the center of mass energy in the range near the resonance is presented. The comparison with a similar effect in these reactions with the neutral $Z$-boson in the intermediate state is given. We show that these effects are $\le 10^{-3}$. The main source of asymmetry is of pure QED origin ($\sim 10^{-2}$) which arises from the interference between initial and final state real photon emission.