Short description of how the implementation will prob go:
in gamestates/game.lua edit game:draw() function so it draws the background before anything else
the same file in game:update(dt) function we will move the background relative to dt (e.g. bgX = bgX + bgVelocity * dt), just like we are moving Player or any other entity in the game
thig might be helpful, a course on love2d, in the linked section they talk about adding a parallax background: https://youtu.be/3IdOCxHGMIo