This is an authentic copy of Ninja Gaiden running on a PC Engine? I've never heard or seen this parallax scrolling issue.
Edit: Holy cow, it really looks to be that awful! I can't believe how awful the scrolling is. Just goes to show patching isnt always such an evil thing, as not all games shipped without issues!
that's running on an emulator, but it behaves the exact same way on my real PC Engine.
Ys 3 has the same sort of parallax scrolling issues. It's doing tile-width scrolling, which is why it looks so off. It's the same way the MSX did a lot of its scrolling. The reason it's scrolling like that is because the PC Engine only had 1 hardware background layer, so to achieve parallax, they needed to offset tiles by the correct amount in the opposite direction to make it look like they were different layers. This is actually a very old trick, they just did it extremely poorly. Manfred Trenz did it to great effect on the C64 in Turrican:
https://www.youtube.com/watch?v=xQVi0k5nUTI#t=16m14s
The reason it works so better here is because he actually animated each tile line by line and would flip through them as they moved until they would reach a full tile width, at which point he'd shift the entire level down. This means each individual tile takes up 8 times amount the normal space (because they are 8 pixels wide, and thus need 8 frames of animation). That's likely why they didn't do this on the PC Engine, because space was at a premium on tiny hucards.