We theoretically study binary mixtures of thin and thick hard rods with diameter ratio more extreme than 1:4. The bulk phase diagram of these systems exhibits a triple point, where an isotropic (I) phase coexists with two nematic phases ( N1 and N2) of different composition. Using density functional theory, we predict that the I-N2 interface is completely wet by N1 upon approach of the the I-N1-N2 triple point. This entropic triple point wetting should be experimentally observable in colloidal suspensions of rodlike particles.