We present radio spectra spanning 0.1-10GHz for the sample of heavily obscured luminous quasars with extremely red mid-infrared-optical colors and compact radio emission. The spectra are constructed from targeted 10GHz observations and archival radio survey data that together yield 6-11 flux-density measurements for each object. Our primary result is that most (62%) of the sample have peaked or curved radio spectra and many (37%) could be classified as Gigahertz-Peaked Spectrum (GPS) sources. This indicates compact emission regions likely arising from recently triggered radio jets. Assuming synchrotron self-absorption (SSA) generates the peaks, we infer compact source sizes (3-100pc) with strong magnetic fields (6-100mG) and young ages (30-104yr). Conversely, free-free absorption (FFA) could also create peaks due to the high column densities associated with the deeply embedded nature of the sample. However, we find no correlations between the existence or frequency of the peaks and any parameters of the MIR emission. The high-frequency spectral indices are steep ({alpha}~-1) and correlate, weakly, with the ratio of MIR photon energy density to magnetic energy density, suggesting that the spectral steepening could arise from inverse Compton scattering off the intense MIR photon field. This study provides a foundation for combining multifrequency and mixed-resolution radio survey data for understanding the impact of young radio jets on the ISM and star-formation rates of their host galaxies. faGithub