The nitrogen-vacancy (NV) center in diamond is known to spin-polarize under optical illumination, but how is this polarization transferred to carbon spins in the bulk of the crystal? Here, the authors use field-cycled nuclear magnetic resonance to study the formation of light-driven ${}^{13}$C spin polarization in NV-rich diamond at room temperature. By considering the system's response to various magnetic fields, it is shown that ${}^{13}$C spins polarize via a cross-relaxation process involving the NV center and the electron and nuclear spins of the substitutional nitrogen impurities. Furthermore, it is shown that high levels of ${}^{13}$C spin polarization can be attained for almost any magnetic field orientation.