Parallax vs tiles is, to me, one of those diminishing returns things. They're both beneficial in opposite situations, and can work together really well. If you've got a lot of locations, if your maps are large, or if the placement is more unique than the content, tiles is generally the way to go. If you've got very different looking locations, a small amount of them, or if the thing being mapped isn't big, then it could be beneficial to do Parallax maps. Parallax mapping allows for more control at the cost of more memory usage, but if you're not making good use of that higher amount of control or you can't afford it taking that much space on disk, then there's not anything gained by using it. Tiles are fantastic for keeping the file size smaller, or being able to reuse the same thing in a different way, but at some point and on some level you will always be able to see the repetition. If not, it shouldn't be done with tiles. If you're doing almost everything unique, tiles will actually get in your way and give you no benefit. In combination, you can get a lot of wow with little memory cost by using parallax for key, unique locations, and spending a little extra love on those tiles for the rest of the game.
That said, I think most people will not get any use out of going full parallax. It requires a certain level of artistry to pull off, and most of the time you can get the same effect in a non-parallax setting. Clever use of self made tiles, using layering to hide repetition, and especially grid free doodads can give you nearly all the benefits with a much lower memory cost.
I should probably add here that as a pixel artist I am biased towards the clever use of tiles. I think they're undervalued and can be so awesome when exploited to the max. Also, when doing pixel art, full parallax maps take a stupidly huge amount of time, because pixel art is one of those diminishing returns things too.