In this paper we study the high-energy exclusive photoproduction of heavy quarkonia pairs in the leading order of the strong coupling constant αs. In the suggested mechanism, the quarkonia pairs are produced with opposite charge parities, and predominantly have oppositely directed transverse momenta. Using the Color Glass Condensate approach, we numerically estimate the production cross sections in the kinematics of the forthcoming electron-proton colliders, as well as proton-ion colliders in ultraperipheral collisions. We find that the cross sections are within the reach of planned experiments and can be measured with reasonable precision. The suggested mechanism has a significantly larger cross section than that of the same C-parity quarkonia pair production.