The results of new spectroscopic analyses of 20 recently reported extrasolar planet parent stars are presented. The companion of one of these stars, HD 10697, has recently been shown to have a mass in the brown dwarf regime; we find [Fe/H]=+0.16 for it. For the remaining sample, we derive [Fe/H] estimates ranging from -0.41 to +0.37, with an average value of +0.18+/-0.19. If we add the 13 stars included in the previous papers of this series and six other stars with companions below the 11 MJ limit from the recent studies of Santos et al., we derive <[Fe/H]>=+0.17+/-0.20. Among the youngest stars with planets with F or G0 spectral types, [Fe/H] is systematically larger than young field stars of the same Galactocentric distance by 0.15 to 0.20 dex. This confirms the recent finding of Laughlin that the most massive stars with planets are systematically more metal-rich than field stars of the same mass. We interpret these trends as supporting a scenario in which these stars accreted high-Z material after their convective envelopes shrunk to near their present masses. Correcting these young star metallicities by 0.15 dex still does not fully account for the difference in mean metallicity between the field stars and the full parent stars sample. The stars with planets appear to have smaller [Na/Fe], [Mg/Fe], and [Al/Fe] values than field dwarfs of the same [Fe/H]. They do not appear to have significantly different values of [O/Fe], [Si/Fe], [Ca/Fe], or [Ti/Fe], though. The claim made in Paper V that stars with planets have low [C/Fe] is found to be spurious, due to unrecognized systematic differences among published studies. When corrected for these differences, they instead display slightly enhanced [C/Fe] (but not significantly so). If these abundance anomalies are due to the accretion of high-Z matter, it must have a composition different from that of the Earth.