Ok. I've run the shader test like 10 times and I did what JibbSmart and painkiller said about isolating the faces. Now I get what Jibb meant about the smooth shades. They vanished after the isolation. The lighting looks pretty good now. In fact, it would be perfect, better yet than xNormal, if it weren't for the darn direction.
After intensively rotating the object in every possible way and position, I came to notice that the parallax left-right scrolling corresponds to tilting instead of panning and vice-versa -- when panning the object, the texture is scrolled vertically.
I took a look inside the .fx and I understand about 10%. I discovered where to edit the strength of the effect but I can't locate the pan and tilt thing, no matter what.

I took the time to study shaders in the past few days and I figured out what to modify in 3DGS default parallax shader. I changed a couple of things in there and it straightened quite a bit. I swapped x and y for the view in tangent space and inverted the heightmap interpretation but it's still buggy:

As you can see, the offsetting looks a little rotated and wobbly (?!). Notice the road wheels, also the narrowing along the bottom of the track and the little wheels from the top.
Would somebody give me a hand, please? Making games was supposed to be fun (well generally, except for making money which is not my case) but frustration took over for the last week because I encountered a problem which letting aside I'm not responsible of, but it's damn hard to correct. So, if somebody would be kind to tell me how is it working on your PC, or if you have a better parallax shader to give away or an idea but the one to let go of parallax.

Another 10 days have past without any solution whatsoever. I couldn't find a working HLSL parallax shader, nor could get to the bottom of the one built-in.
What's up with this mystery around the parallax shader? It's like some tabu thing that makes everybody uncomfortable. I really don't get it.

Anybody who knows shaders or really anybody at all, can help me get out of this one?