A conservative constraint on the rest mass of the photon can be estimated under the assumption that the frequency dependence of dispersion from astronomical sources is mainly contributed by the nonzero photon mass effect. Photon mass limits have been set earlier through the optical emissions of the Crab Nebula pulsar, but we demonstrate that these limits can be significantly improved with the dispersion measure (DM) measurements of radio pulsars in the Large and Small Magellanic Clouds. The combination of DM measurements of pulsars and distances of the Magellanic Clouds provides a strict upper limit on the photon mass as low as m(gamma) <= 2.0 x 10(-45) g, which is at least four orders of magnitude smaller than the constraint from the Crab Nebula pulsar. Although our limit is not as tight as the current best result (similar to 10(-47) g) from a fast radio burst (FRB 150418) at a cosmological distance, the cosmological origin of FRB 150418 remains under debate; and our limit can reach the same high precision of FRB 150418 when it has an extragalactic origin (similar to 10(-45) g).