We surveyed a sample of compact Galactic planetary nebulae (PNe) with the Space Telescope Imaging Spectrograph on the Hubble Space Telescope (HST) to determine their gas-phase carbon abundances. Carbon abundances in PNe constrain the nature of their asymptotic giant branch (AGB) progenitors, as well as cosmic recycling. We measured the carbon abundances, or the limits thereof, of 11 compact Galactic PNe, notably increasing the sample of Galactic PNe whose carbon abundance based on HST ultraviolet spectra is available. The dust content of most targets has been studied elsewhere from Spitzer spectroscopy; given the compact nature of the nebulae, both UV and IR spectra can be directly compared to study gas- and dust-phase carbon. We found that carbon-poor (C/O < 1) compact Galactic PNe have an oxygen-rich dust type, while their carbon-enhanced counterparts (C/O > 1) have carbon-rich dust, confirming the correlation between gas- and dust-phase carbon content that was known for Magellanic Cloud PNe. Based on models of expected final yields from AGB evolution, we interpret the majority of the carbon-poor PNe in this study as the progeny of ~1.1-1.2 M ⊙ stars that experienced some extra mixing on the red giant branch. They went through the AGB but did not go through the carbon star phase. Most PNe in this group have a bipolar morphology, possibly due to the presence of a subsolar companion. The carbon-enhanced PNe in our sample could be the progeny of stars in the ~1.5-2.5 M ⊙ range, depending on their original metallicity.