I guess the big 16x16 sprites of the PC Engine and Neo Geo makes it easier to make parallax scrolling or other split-screen effects. I read somewhere that Hudson planned for two background layers first, but I guess it was already expensive as it was so they had to scrap it.
Dual tile layers are used in many ways, intended for many different things and are not simply a "hardware parallax" feature. What they are often used for is faking large sprites for things like bosses, when the PC Engine usually uses real actual sprites. Are those fake sprites? Is that hardware support for sprites? Can tiles be sprites, even though it's being insisted that the opposite doesn't count?
In colloquial terms it would probably be considered a sprite, but for "sprite" as a hardware term it's obviously not a sprite, so yeah call it "fake" or whatever.
Using multiple hardware tile layers for parallax is no different than using multiple hardware sprites for parallax or a single hardware tile layer for parallax or a combination of hardware tiles and sprites for parallax. In all cases, parallax is not the dedicated function of the hardware features used and all features used are hardware supported.
I see your point, parallax scrolling (unlike sprites and background layers) isn't a hardware feature in its own in any of these systems, and it was a thing long before multiple background layers became common and made it easy to do.
"Hardware parallax" may be a clumsy expression, although in the context I thought it was clear what Artabasdos meant.