We investigate the electronic band structure of graphene on a series of two-dimensional magnetic transition -metal phosphorus trichalcogenide monolayers, MPX3 with M = {Mn,Fe,Ni,Co} and X = {S,Se}, with first-principles calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and sublattice resolved proximity-induced exchange couplings (lambda(A)(ex) and lambda(B)(ex)) from the low-energy Dirac bands of the proximitized graphene. Depending on the magnetic phase of the MPX3 layer (ferromagnetic and three antiferromagnetic ones), completely different Dirac dispersions can be realized with exchange splittings ranging from 0 to 10 meV. Remarkably, not only the magnitude of the exchange couplings depends on the magnetic phase, but also the global sign and the type. Important, one can realize uniform (lambda(A)(ex) approximate to lambda(B)(ex)) and staggered (lambda(A)(ex) approximate to -lambda(B)(ex)) exchange couplings in graphene. From selected cases we find that the interlayer distance, as well as a transverse electric field, are efficient tuning knobs for the exchange splittings of the Dirac bands. More specifically, decreasing the interlayer distance by only about 10%, a giant fivefold enhancement of proximity exchange is found, while applying few V/nm of electric field, provides tunability of proximity exchange by tens of percent. We have also studied the dependence on the Hubbard U parameter and find it to be weak. Moreover, we find that the effect of SOC on the proximitized Dirac dispersion is negligible compared to the exchange coupling.