It doesn’t look like any parishes have been mapped in New Brunswick. There are counties (admin_level=6) and municipalities (admin_level=8), but no parishes (which would presumably get tagged with admin_level=7).
It isn’t clear whether the parishes should even be in OSM anyway. According to this article:
AFAIK, census areas aren’t typically added to OSM. Since these areas no longer have any other significance, they probably shouldn’t be in OSM at all.