Precise trigonometric parallax measurements orders of magnitude more accurate than a milliarcsecond will contribute greatly to our understanding of peculiar, low luminosity stellar objects of several types. First, the volume of space out to which luminosities may be determined to the accuracy of the best, very-nearby stars will be increased greatly. For the relatively rare field Population II stars, this will lead to the first accurate empirical calibrations of the main sequence at the low mass end, for comparison with globular clusters of various metallicities. Parallaxes at 1 kpc or farther will be adequate to help in the discovery or confirmation of the rare carbon dwarfs - main sequence stars with carbon-rich atmospheres. For cool white dwarfs, luminosities accurate to a few per cent or better will identify unresolved binaries, and objects of unusually high and low mass. For our most numerous solar neighbors, the M dwarfs and especially those near the stellar mass limit, accurate luminosities can help in the determinations of the chemical composition and age distributions.