What size of background should I make for my fairly big scene?

I have been trying to look for an answer for a while. I couldn't find a clear answer to the following concern, or I lost my search skills :)

I'm currently working on a 2D platformer side-scroller game on PC. It looks somehow close to the game "Dust: An Elysian Tail", just to give a clear reference on how the game scenes will be.

My concern is regarding the backgrounds in the scene. I'm planning to use 3 layers of backgrounds to achieve parallax effect. I couldn't find clear information about how big, in pixels, should we draw them to cover all the game-play area in the scene. My scenes are fairly big and they have lots of upward and downward paths. So it becomes a huge image, if I try to cover the whole scene. I believe this will affect the memory too. I don't want to do seamless backgrounds and connect them together as they will not look nice and cool. I want the background to be beautiful, unique and high quality.

I thought of drawing the backgrounds in 4k and resize them how I like in unity using the transform component. Is it a good approach to do it this way? Will I lose quality if I resize my backgrounds? Is there a better way to approach this?

Appreciate you help :)