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.