What Tolkien Officially Said About Elf Sex (2002)
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I can totally imagine a future civilization taking LotR or ASOIAF as the basis for a religion. Not that it would be a world I'd want to live in, but I can see it. If it can happen for Scientology, anything is possible.
Motivation may differ though, intellectual curiosity in one case, fear of eternal damnation in another.
Writing about moral principles, the struggle between good and evil, how greed can corrupt, etc, isn't the same as writing a "parallel" to Christianity. Those themes are universal to humankind, not specific to a single religion.
But to be honest, as characters the Dresden Fae makes sense. And humans can and do become Fae, if only at the cost of your mortal soul. So not that much separation unlike the Firstborn and the Atani.
(Incidentally, I really like this fanfic about Harry Potter waking up in Dresdenverse and got hitched with Aurora, the Seelie Summer Lady; breaking the canon plot in the process. I don't think a similar relationship can really happen between a Tolkien Elf and a human (or other mortal) without something really special going on. Like Aragorn or, uh, his First Age ancestors.