... is it the parent who brought that child into said situation with the full awareness that adults use that public place also, and thus adult language is only natural?
In my opinion cursing is not "adult language", it is foul language. There may be situations were it is understandable that people curse, but most of the times, in such situations with children when parents reprimand others, there was no overall need to curse.
As others stated, you would not curse in a theater or a church, so I see no reason to curse in open public at all. If you are in a bar, a club or places were it is okay to drink alcohol and such, I personally have nothing against it, because these places can be termed as "foul", so one expects this language.
I once played a game with an online chat, which had a french and an english chat. I usually stayed in the french chat, although I rarely understood anything, but there was just too much foul language on the english one, most of the time with no apparent reason. I dislike being around rude and disrespectful people.