Find an entry's parent category if it's not explicitly assigned

In the context of an entry archive, if an entry has a subcategory as its primary category, but is not explicitly assigned to that subcat's parent category, how can I get the label of that parent? I figured it out for the category archive itself, but have hit a wall on the entry archive.

I'm using top-level categories as the main nav, so want to use the top-level category basename to set the active state in the nav, but would like to avoid requiring the authors to set the top level category, if possible.