We present a simple treatment of the depletion-driven demixing transition in isotropic and perfectly aligned binary hard-rod mixtures. The simplicity is mainly due to the fact that we use the second virial approximation to the Helmholtz free energy in combination with a convenient Legendre transform. This combination leads to an exact expression for the depletion-driven demixing spinodal in isotropic mixtures of hard rods with different diameters. We show for rod species with the same length that the demixing spinodal is thermodynamically stable with respect to the isotropic-nematic transition, if the diameter ratio is larger than about 5. We also show that perfectly aligned rod mixtures show a nematic-nematic demixing spinodal, that may preempt the nematic-smectic or the nematic-columnar bifurcation if the size difference between the particles is sufficiently large.