The phase behavior of symmetric binary rod-plate mixtures has been investigated by numerical minimization of a free energy functional derived by Parsons [J. D. Parsons, Phys. Rev. A 19, 1225 (1979)] and Lee [S. D. Lee, J. Chem. Phys. 87, 4972 (1987)]. Both rod and plate molecules are represented by hard cylinders, with aspect ratios chosen so that the molecular and pair excluded volumes are equal; in this way symmetric phase diagrams in composition are found. The subtle competition between the packing entropy and the entropy of mixing rules out the possibility of a uniaxial nematic-biaxial nematic phase transition and instead favors a demixing phase transition between a rod-rich and a plate-rich nematic phase. It is shown that the biaxial nematic phase is unstable relative to demixing even for symmetric mixtures of very long rod and very flat plates, where the Parsons-Lee theory becomes identical with the Onsager theory. The contradictory predictions obtained in recent studies regarding the stability of the biaxial nematic phase have been resolved by examining the lowest aspect ratio of the rods (kappa(2)) where the Parsons-Lee and Onsager theories become equivalent. It turns out that neglecting the lower order terms in the excluded volumes (so-called end effects) leads to a favoring of the biaxial nematic phases. Only two types of phase transitions are observed in this work: isotropic-nematic phase coexistence and demixing transitions involving either two isotropic or two nematic phases. The stability of the nematic region on mixing is found to be very sensitive to the aspect ratios of the molecules: for moderate aspect ratios of the rods (510 the opposite tendency is found. A demixing transition between rod-rich and plate-rich nematic phases takes place for all of the systems studied with a widening coexistence region for increasing aspect ratio. Isotropic-isotropic demixing is also observed for aspect ratios kappa(2)>65.5. For the larger values of kappa(2) the regions of isotropic-isotropic and nematic-nematic demixing broaden, while the isotropic-nematic coexistence is progressively suppressed.