Nominatim prefers to associate streets with administrative boundaries that they are inside of. The data for counties is pretty solid, and in the U.S., pretty much every street is properly inside a county (or equivalent). The data for other administrative units is less solid (and often times less clear in the real world). If a street is not found inside of a city or town or the like, it is assigned to the nearest "place" node (I think with some accommodation for significance of the place). So, for instance, that's how Blackgum Drive ends up assigned to Marblehill.
I'm not sure how other people are thinking about it, but I think the long term solution to what you are seeing is to build up exact address information and then also index that data for street-city (or place) relationships in search results. Given the way zip codes work and the hand waviness of postal addressing (the post office does a good job correctly routing miserably addresses mail), this is probably more sensible that trying to create boundaries.
It would also be possible to index streets to several nearby places (but of course that's more work up front and more data to sift through on each search).
http://www.openstreetmap.org/way/9428636
And it is not located in Jasper:
http://www.openstreetmap.org/relation/119636
So is this the right street? Or should the street be where you linked the map? Because I can't find a street named Disharoon Dr. there.