The flickering Mario and Mario moving horizontally is just weird ‒ especially because why does Mario even move automatically foreward in the first palce? That needs to be looked with a debugger so you may want to send me a BPS and presumably all necessary files.
As for the broken parallax, I wrote in the readme that a sprite status bar is recommend because
a) the background uses parallax HDMA and
b) overwrites the status bar colours
I also wroe that a layer 3 status bar might work but I wrote there that you need to do some tweaks. The first thing is that shifting the tilemap isn't enough because the HDMA isn't dependent on it (HDMA doesn't know what a tilemap is so if you modify the tilemap, it doesn't shift automatically downwards) and have to adjust the HDMA table manually. In that case, I did that part myself so no need to fillde with it (untested).
Another issue is the palette which, again, might work if you use either seperate status bar colours or modify the tilemap so that the background uses the first half of palette 0 and 1.
Okay, my layout looks ugly.