The occurrence of proton tunneling in MAPbI hybrid organic inorganic perovskites is demonstrated through the effect of isotopic labeling of the methylammonium (MA) component on the dielectric permittivity response. Deuteration of the ammonium group results in the acceleration of proton migration (inverse primary isotope effect), whereas deuteration of the methyl group induces a normal secondary isotope effect. The activation energies for proton migration are calculated to be 50 and 27 meV for the tetragonal and orthorhombic phases, respectively, which decrease upon deuteration of the ammonium group. The low activation barrier and the deviation from unity of the ratio of the pre-exponential factors (A/A = 0.3-0.4) are consistent with a tunneling mechanism for proton migration. Deuteration of the PEDOT:PSS hole transport layer results in a behavior that is intermediate between that of the deuterated and undeuterated perovskite, due to extrinsic ion migration between the two materials.