Wording "ir de paseo a través la calle" - obviously can be understood especially in context, but does not sound natural/right. Prepositional phrase can be rather "a través de", because simple "a través" do exist too but has meaning of direction in which you are moving ("En dirección transversal"). You should think of "pasear por" as fixed phrase which translates into stroll/walk in some place/through some place, and used to talk about walking for pleasure, while "a través de" more conveys an idea of crossing some area from one side to another rather than strolling/walking through it for pleasure (it that helps think of "navegar por el oceano" y "travesar el oceano" - meaning is quite different). So if you want to stick with "paseo" again your preposition should still be "por" and verb which is used with paseo in the sense you need is "dar un paseo", so it should be "DAR un paseo POR la calle". As paseo already contains idea of ir in a way, it is redundant to say "ir de paseo" and "dar un paseo" what is normally used (compare with something like "to have/take a stroll" which I guess sounds better than "go/walk for a stroll").