Nonlinear photocurrent responses generally require inversion-symmetry breaking, but its origin can be magnetic rather than structural. In this study, we investigate the bulk photovoltaic effect in antiferromagnetic MnS2 and altermagnetic MnSe2, two pyrite-type semiconductors whose magnetic ordering breaks inversion symmetry P and its combination with time reversal PT, while the accompanying lattice distortion is minute. We calculate the shift current (SC) and injection current (IC) responses under linearly and circularly polarized light with and without spin-orbit coupling (SOC) and identify their symmetry-allowed tensor components using the corresponding magnetic and spin point groups, respectively. Linear SC and circular IC are symmetry-allowed in both MnS2 and MnSe2 irrespective of SOC, whereas linear IC and circular SC remain forbidden in MnS2 but become allowed in MnSe2 only when SOC is included. These results highlight the importance of SOC and the corresponding MPG analysis for a complete description of nonlinear charge photocurrents in magnetic materials. Beyond charge responses, MnSe2 supports linear shift and circular injection spin photocurrents in the nonrelativistic limit, whereas all spin-photocurrent responses are forbidden in MnS2. Our findings therefore establish inversion symmetry breaking induced by compensated magnetic order as a route to generating and controlling nonlinear photocurrents.