We examine the quenched fraction of central and satellite galaxies as a function of galaxy stellar mass, halo mass, and the matter density of their large-scale environment. Matter densities are inferred from our ELUCID simulation, a constrained simulation of the local universe sampled by SDSS, while halo masses and central/satellite classification are taken from the galaxy group catalog of Yang et al. The quenched fraction for the total population increases systematically with the three quantities. We find that the"environmental quenching efficiency," which quantifies the quenched fraction as a function of halo mass, is independent of stellar mass. And this independence is the origin of the stellar mass independence of density-based quenching efficiency found in previous studies. Considering centrals and satellites separately, we find that the two populations follow similar correlations of quenching efficiency with halo mass and stellar mass, suggesting that they have experienced similar quenching processes in their host halo. We demonstrate that satellite quenching alone cannot account for the environmental quenching efficiency of the total galaxy population, and that the difference between the two populations found previously arises mainly from the fact that centrals and satellites of the same stellar mass reside, on average, in halos of different mass. After removing these effects of halo mass and stellar mass, there remains a weak, but significant, residual dependence on environmental density, which is eliminated when halo assembly bias is taken into account. Our results therefore indicate that halo mass is the prime environmental parameter that regulates the quenching of both centrals and satellites.