You should cut a large background into equal smaller ones.
You most likely do not need a tile map to scroll vertically. Just add your smaller backgrounds to the array and show only those that are visible to the player.
In this example, I am updating only parts of BG1 , BG2 and BG3 . When BG1 turns off the screen, I will remove it and add BG4 above BG3 . The same thing happens with BG5 , BG6 , etc.

You can also create an infinitely looped background by sorting an array with smaller parts of the background.
The use of parallax scrolling depends on your game art. The parallax effect is created by two or more background layers that scroll at different speeds. Watch a few videos to better understand it.
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