  We present radio spectra spanning $0.1 - 10$ GHz 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 10 GHz
observations and archival radio survey data, which together yield $6-11$ flux
density measurements for each object. Our suite of Python tools for modeling
the radio spectra is publicly available on Github. Our primary result is that
most (61%) of the sample have peaked or curved radio spectra and many (36%)
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 ($3 - 100$ pc) with strong magnetic fields ($6 - 100$ mG)
and young ages ($30 - 10^4$ years). 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 ($\alpha \approx -1$) 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 multi-frequency 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.
