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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.).

However, my 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 平行 encompass 並行 like 大辞泉 says?

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When we say 平行線, it's more common to write 平行線 than 並行線. But by definition of 大辞泉 and 大辞林, they seem to have the same meaning as you point out. They are not clearly distinguished even by most Japanese in everyday life and they would choose 平行 over 並行 for its familiarity. In the philosophical and psychological term 「並行論」 (parallelism), it seems 平行 is not used in 大辞泉 and 大辞林 ( dic.yahoo.co.jp/… ) So there could be some differences but when I search 平行論 in Google there are some hits on it. –  Teno Oct 1 '12 at 17:50

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up vote 7 down vote accepted

平行 can mean the same thing as 並行 in the second sense (物事が同時に行われる), so it does not encompass 並行 completely. For example, 電車と並行して走る should not use 平行 (although don't be surprised if you see the two mixed up). However, just look at the characters, the 並 of 並行 can be seen in words like 並ぶ while the 平 of 平行 can be seen in words like 平面. So, in general, 平行 is primarily used in the mathematical sense while 並行 is used to just mean "do two things at once" or "line up and proceed together" like in 英語とフランス語を並行して勉強する . Also, notice that parallel in the computing sense is 並列, not 並行, which would be concurrent computing.

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プログレッシブ says: "並行処理|〔コンピュータ〕parallel processing" –  istrasci Oct 1 '12 at 22:18
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@istrasci: That dictionary is wrong. Here are a few links: 1, 2, 3. –  Jesse Good Oct 1 '12 at 23:39
    
It's built into Mac OSX. But I think that might be the result of the non-computer-sciency-savvy layperson's confusion between "concurrent" and "parallel". –  istrasci Oct 2 '12 at 1:38
    
@istrasci: Yes, you are probably right, as in everyday speech the meanings are quite similar. –  Jesse Good Oct 2 '12 at 2:14

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