This post maybe should be considered an epilogue to my five-part series "The New Cleric is the Old Cleric", but you'll just have to take it for what it is. By the time I was done writing that series, one of the things I ended up convincing myself of was that priests did not have to be clerics, nor did clerics have to be priests, in 4th Edition Dungeons & Dragons. This is because most of the non-combat 'magic' that might be relevant to parish priests could be thought of as ritualistic magic brought about by ceremony and tradition - not necessarily by fiery innate powers handed down by the gods on high. I've said it before, and I'll say it again, but one of the things I love about 4E is that Ritual Magic is available to any character class. This is why parish priests can easily be any class without necessarily breaking the RAW. All a character needs is skill in Religion and the Ritual Caster feat and they qualify.
So, two possibilities jump to mind. The first of which is that each of the core religions and deities presented in the 4E Players Handbook and Dungeon Masters Guide could very well have legions of 'priests' who were fighters, rangers, rogues, warlocks, etc - depending on the tenets of their deity.
It could be that the priests of...
... Asmodeus are Evil warlocks and warlords.
... Avandra are Good bards and rogues.
... Bahamut are Lawful Good paladins and clerics.
... Bane are Evil fighters and paladins.
... Corellon are bards and rangers.
... Erathus are warlords and wizards.
... Gruumsh are Chaotic Evil barbarians and rogues.
... Ioun are artificers and wizards.
... Kord are barbarians and fighters.
... Lolth are rogues and warlocks.
... Melora are bards and rangers.
... Moradin are Lawful Good artificers and paladins.
... Pelor are Good paladins and rangers.
... The Raven Queen are clerics and wizards.
... Sehanine are bards and warlocks.
... Tharizdun are Chaotic Evil artificers and warlocks.
... Tiamat are Evil fighters and rogues.
... Torog are Evil fighters and wizards.
... Vecna are Evil clerics and wizards.
... Zehir are Evil rogues and warlocks.
Of course, the previous list are only suggestions of classes for parish priests. Any character class, NPC, or monster could act be a priest for any of these deities under the right circumstances. They should all have, however, two things in common: skill training in Religion and the Ritual Casting feat. If Religion is not available to them as a class skill, then the character should also choose the Skill Training (Religion) feat at first level. If the Ritual Casting feat is not a class feature for the character, then this feat should also be taken at first (or second) level. By second level, all characters who are parish priests should have both Religion skill training and the Ritual Casting feat. Some religions may also require additional skill training in Arcana, Healing, History, or Nature so that the appropriate rituals would be made available to the parish priest.
The second possibility that comes to mind is that the parish priest concept could easily be fully fleshed out as a multiclass character type - perhaps as a "Multiclass Priest"character type like the excellently written Multiclass Noble that is produced by Aleagames. Maybe if an artist out there wants to join forces with me, we could put something like this together... there are all kinds of cool 'powers' you could think of for a multiclass parish priest class...
Confessions of the Knife (standard; daily) • Divine, Implement
Close Burst 5, [Cha] vs. Will; the target must immediately drop all unconcealed weapons. Failure: the target is overcome with guilt and suffers a -5 penalty on attack rolls until the end of their next turn.