The Very Long Baseline Array has been used at 15 GHz to image the milliarcsecond structure of the Seyfert galaxies Mrk 231 and Mrk 348 at two epochs separated by about 1.7 yr. Both galaxies contain parsec-scale double radio sources whose components have brightness temperatures of 10^9-10^{11} K, implying that they are generated by synchrotron emission. The nuclear components are identified by their strong variability between epochs, indicating that the double sources represent apparently one-sided jets. Relative component speeds are measured to be ~0.1c at separations of 1.1 pc or less (for H_0 = 65 km/s/Mpc), implying that parsec-scale Seyfert jets are intrinsically different from those in most powerful radio galaxies and quasars. The lack of observed counterjets is most likely due to free-free absorption by torus gas, with an ionized density n_e > 2 X 10^5 cm^{-3} at T~8000 K, or n_e > 10^7 cm^{-3} at T~10^{6.6} K, in the inner parsec of each galaxy. The lower density is consistent with values found from X-ray absorption measurements, while the higher temperature and density are consistent with direct radio imaging of the NGC 1068 torus by Gallimore et al.
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