Near-field thermal radiation mediated by surface plasmons in parallel graphene nanodisk dimers is studied using a semianalytical model under the electrostatic approximation. The radiative heat transfer between two disks as a function of the distance between them in coaxial and coplanar configurations is first considered. Three regimes are identified and their extents determined using nondimensional analysis. When the edge-to-edge separation is smaller than the disk diameter, near-field coupling and surface plasmon hybridization lead to an enhancement of the radiative heat transfer by up to four orders of magnitude compared to the Planck blackbody limit. A mismatch in the disk diameters affects the plasmonic mode hybridization and can either diminish or enhance the near-field radiation. Destructive interference between eigenmodes that emerge when the relative orientation between disks is varied can induce a twofold reduction in the radiative heat transfer. In all configurations, the radiative heat transfer properties can be controlled by tuning the disk size/orientation, the substrate optical properties, and graphene's doping concentration and electron mobility.