They both seem to mean "parallel". In my
プログレッシブ dictionary, it seems to point to the former being the more mathematical sense of the definition (two co-planar, non-intersecting lines), while the latter indicates a more metaphorical definition (doing things at the same time, parallel computing, etc.).
大辞泉 has the following:
平行：「並行（へいこう）②」に同じ。（並行② is the metaphorical definition)
So which is it? Are they distinct like
プログレッシブ says, or does