The naming of _parallel_fit_estimator does not reflect the serial/single-threaded operation actually performed by the method. No parallel calls are made in the method. The parallel part will come when we pass this method to the Parallel class, as we do here and here.

In PR #16539, it was suggested that we rename this method to _fit_single_estimator to clarify the usage intent. See this comment specifically: https://github.com/scikit-learn/scikit-learn/pull/16539/files#r385937964