Hello Dream Data Dev. I'm making a Yume Nikki fan project in Unity as well, and I was wondering how you made the looping worlds. Hope this isn't too much of a hassle!
I just make sure both sides look identical, and then teleport the player and camera to the other side when they leave bounds. To make sure it doesn't look like it stutters, I add the position offset back on instead of just jumping to set coordinates. For example, I might set the horizontal boundaries as -50 and 50. I continuously check if the player character leaves those bounds. If he does, like if his x position is now 50.4, I get the offset from 50, which is 0.4, and I add that to the left bound. -50 + 0.4 = -49.6. So I change the player x position to -49.6.
As for the parallax background, I have a tessellating texture plane as a child to the camera. I adjust the offset of its material by multiplying the camera's position by some factor.