Background
With the normal parallax effect in Godot (parallaxLayer node) the background can shift from left to right. However, the level design will allow for the player to move up and down as well, so we need to write a script to make the background follow the player nicely. An extreme but great example of this is in Hollow Knight (link: https://www.youtube.com/watch?v=0QnLG-JEzzU&ab_channel=MonStahLer, here you can see when the Hollow Knight goes up and down the background loosely follows the player).
AC
If you want a (bad) example of this logic, here's the code to my old game doing this: https://github.com/TSherpa10/Brave-Summit/blob/main/scenes/background.gd
I think I want slight improvement to this logic.