We investigate the photon-mediated thermal transport between a superconducting electrode and a normal metal. When the quasiparticle contribution can be neglected, the photon-mediated channel becomes an efficient heat transport relaxation process for the superconductor at low temperatures, being larger than the intrinsic contribution due to the electron-phonon interaction. Furthermore, the superconductor-normal metal system acts as a nearly perfect thermal diode, with a rectification factor up to 108 for a realistic aluminum superconductor. The rectification factor can also be tuned in a phase-controlled fashion through a non-galvanic coupling, realized by changing the magnetic flux piercing a superconducting quantum interference device, which modifies the coupling impedance between the superconductor and the normal metal. The scheme can be exploited for passive cooling in superconducting quantum circuits by transferring heat toward normal metallic pads where it dissipates more efficiently or for more general thermal management purposes.