At first I thought Parrish was a phoenix but then I started to expand my thinking on this past just that. So then I looked at a Griffin but after reading on them I saw that they were most used in guarding treasures which I don't see on Teen wolf. Then I came across the Hell Hound and this makes the most sense to me.
1. A Hell Hound is Fire based like Parrish and i'm sure could survive being burned by it as well.
2. Not sure if anyone has seen the episodes of the show Supernatural that showed Heel hounds, but Hell Hounds are very good at masking their presence and were unable to be seen unless a spell was cast to be able to see them. Now even though Scott along with his pack and Derek could physically see Parrish they couldn't see any aura around him like they could with Kira and her Kitsune powers nor tell what he was. Maybe this is a different way in showing us the same things that happened on supernatural but in a Teen wolf style.
3. A Hell hound was known for guiding lost souls back home after death. Now maybe this is why we are seeing Parrish taking the dead bodies, he's guiding their supernatural souls back home, and when I say home i'm referring to the Nemeton the thing that we know draws supernatural creature to Beacon hills. Perhaps just perhaps Parrish is a Hell hound creature that works specifically for carrying souls to the Nemeton which is probably some form of resting place for supernatural creatures.
This is just my theories on this I could be wrong but I think i'm on to something here.
Richard Morgan/ Hybrid Klaus out