We show that p-wave magnets efficiently generate magnetization via a linear thermal Edelstein effect arising from the orbital magnetic moments of their magnons. Furthermore, they can generate perfectly nonreciprocal orbital currents through a purely even-order nonlinear response. This makes them promising candidates for orbital-current rectification. Because these transport phenomena originate from the orbital magnetic moment of magnons, they connect magnonics and orbitronics. More generally, we find that odd-parity-wave magnets with coplanar ground states host magnons with zero spin magnetic moment. Instead the magnons carry a collinear, out-of-plane orbital magnetic moment, realizing an orbital version of antialtermagnetism. We establish these general results using symmetry arguments and demonstrate them explicitly for a coplanar ground state with the minimal number of sublattices, inspired by the ground state of CeNiAsO. Our conclusions hold both in absence and presence of spin-orbit coupling.