Atmospheric retrievals are now a standard tool to analyze observations of exoplanet atmospheres. This data-driven approach quantitatively compares atmospheric models to observations in order to estimate atmospheric properties and their uncertainties. In this paper, we introduce a new retrieval package, the PHOENIX Exoplanet Retrieval Algorithm (PETRA). PETRA places the PHOENIX atmosphere model in a retrieval framework, allowing us to combine the strengths of a well-tested and widely-used atmosphere model with the advantages of retrieval algorithms. We validate PETRA by retrieving on simulated data for which the true atmospheric state is known. We also show that PETRA can successfully reproduce results from previously published retrievals of WASP-43b and HD 209458b. For the WASP-43b results, we show the effect that different line lists and line profile treatments have on the retrieved atmospheric properties. Lastly, we describe a novel technique for retrieving the temperature structure and e− density in ultrahot Jupiters using H− opacity, allowing us to probe atmospheres devoid of most molecular features with the James Webb Space Telescope.