We have discovered two low-ionization broad absorption line (BAL) quasars in programs to obtain optical spectra for radio-selected quasar candidates from the VLA Faint Images of the Radio Sky at Twenty Centimeters (FIRST) Survey. Both belong to the extremely rare class of BAL QSOs that exhibit narrow absorption lines from metastable excited levels of Fe II and Fe III. Until now, there was just a single object in this class, 0059-2735. In addition, one of our new objects is the first known radio-loud BAL QSO. The properties of these three unusual objects suggest a trend of increasing radio luminosity with the amount of absorption to the quasar, and are perhaps transition objects between radio-loud and radio-quiet quasars. The two new objects are from a radio-selected sample comprising less than 200 quasars; one is heavily attenuated at optical wavelengths in the observed frame. These objects would be easily overlooked by most optical QSO searches; their abundance in the radio sample suggests that they may be representatives of a largely undetected component of the quasar population, perhaps as numerous as ordinary low-ionization BAL QSOs, which constitute 1%-2% of all QSOs.