If you're the only game in town, you probably do accommodate just about everyone. The most common places I've seen fairly universal coverage in the USA is in hospital chapels and in chapels located on military bases.
Also, let's be clear about the definition of "parish".* Over here, the usual church definition of "parish" is a geographical area in which all members of the faith are served by a particular church. You don't necessarily have to be a regular churchgoer to be counted, they'll also include those who are occasional attendees. They will also accommodate out-of-towers if at least one of the persons involved is also a fellow believer. And sometimes, they check. But you probably should be registered if you want to avail yourself of their services.
And I'm pretty sure that's not a uniquely American attitude. I've seen the same position struck by clergymen in UK fiction more than once. While they could just be making it up for the show, somehow, I don't think so.
* BTW, in Louisiana, the word is ALSO used as a synonym for "county"