An integral equation treatment of connectedness percolation in macromolecular fluids is extended to athermal mixtures of semiflexible and rodlike polymers. The percolation threshold for formation of a geometrically connected network of rodlike particles dispersed in a medium of flexible polymers is investigated as a function of the particle aspect ratio. Results are also presented for a one-component model of identical rodlike particles in the absence of a matrix. The dependence of critical volume fraction on rod aspect ratio is strikingly similar in both instances, and close to that predicted by an analytic thread-chain approximation. The primary effect of explicitly including the athermal, flexible, matrix polymer molecules is a reduction of the critical volume fraction by a factor which depends only weakly on the aspect ratio of the rodlike particles.