We present new subarcsecond-resolution Karl G. Jansky Very Large Array (VLA) imaging at 10 GHz of 155 ultraluminous (L bol ∼ 1011.7-1014.2 L o˙) and heavily obscured quasars with redshifts z ∼ 0.4-3. The sample was selected to have extremely red mid-infrared-optical color ratios based on data from the Wide-Field Infrared Survey Explorer (WISE) along with a detection of bright, unresolved radio emission from the NRAO VLA Sky Survey (NVSS) or Faint Images of the Radio Sky at Twenty cm Survey. Our high-resolution VLA observations have revealed that the majority of the sources in our sample (93 out of 155) are compact on angular scales <0.″2 (≤1.7 kpc at z ∼ 2). The radio luminosities, linear extents, and lobe pressures of our sources are similar to young radio active galactic nuclei (e.g., gigahertz-peaked spectrum [GPS] and compact steep-spectrum [CSS] sources), but their space density is considerably lower. Application of a simple adiabatic lobe expansion model suggests relatively young dynamical ages (∼104-7 yr), relatively high ambient ISM densities (∼1-104 cm-3), and modest lobe expansion speeds (∼30-10,000 km s-1). Thus, we find our sources to be consistent with a population of newly triggered, young jets caught in a unique evolutionary stage in which they still reside within the dense gas reservoirs of their hosts. Based on their radio luminosity function and dynamical ages, we estimate that only ∼20% of classical large-scale FR I/II radio galaxies could have evolved directly from these objects. We speculate that the WISE-NVSS sources might first become GPS or CSS sources, of which some might ultimately evolve into larger radio galaxies.