We show that the luminosity dependence of the red clump stars on age and metallicity can cause a difference of up to < ~0.6 mag in the mean absolute I magnitude of the red clump between different stellar populations. We show that this effect may resolve the apparent ~0.4 mag discrepancy between red clump-derived distance moduli to the Magellanic Clouds and those from, e.g., Cepheid variables. Taking into account the population effects on red clump luminosity, we determine a distance modulus to the LMC of 18.36 +/- 0.17 mag, and to the SMC of 18.82 +/- 0.20 mag. Our alternate red clump LMC distance is consistent with the value (m-M){LMC} = 18.50 +/- 0.10 adopted by the HST Cepheid Key Project. We briefly examine model predictions of red clump luminosity, and find that variations in helium abundance and core mass could bring the Clouds closer by some 0.10--0.15 mag, but not by the ~0.4 mag that would result from setting the mean absolute I-magnitude of the Cloud red clumps equal to the that of the Solar neighborhood red clump. Comment: Accepted for publication in The Astrophysical Journal Letters, AASTeX 4.0, 10 pages, 1 postscript figure