One of the most remarkable properties of the nitrogen-vacancy (NV) center in diamond is that optical illumination initializes its electronic spin almost completely, a feature that can be exploited to polarize other spin species in their proximity. Here we use field-cycled nuclear magnetic resonance (NMR) to investigate the mechanisms of spin polarization transfer from NVs to 13C spins in diamond at room temperature. We focus on the dynamics near 51 mT, where a fortuitous combination of energy matching conditions between electron and nuclear spin levels gives rise to alternative polarization transfer channels. By monitoring the 13C spin polarization as a function of the applied magnetic field, we show 13C spin pumping takes place via a multi-spin cross relaxation process involving the NV- spin and the electronic and nuclear spins of neighboring P1 centers. Further, we find that this mechanism is insensitive to the crystal orientation relative to the magnetic field, although the absolute level of 13C polarization - reaching up to ~3% under optimal conditions - can vary substantially depending on the interplay between optical pumping efficiency, photo-generated carriers, and laser-induced heating.