THIS CAN BE MISSED.
The Dimension is located at Promonotory Island near the end of the story. The dragon will fly you to this island after your second trip to the ice castle. To the west of the starting point is a fork to the north and south. Go north and you find a green chili in a chest. The parallel dimension is in a hidden path to the right of that chest.
Go into the passage to the right as far as you can, then do the following:
Down 1, Right 2, Up 2, Right 2, up 2, left 1, up 2, right 1, up 2, right 2, down 2, right 1, down 4, then just go right. (Credit to shawn_fryer on Steam for these directions)
Please keep in mind that the screen does not scroll every time you step, and there is no way of knowing whether you moved every time other than just trusting the directions.
This is easier to follow if you walk instead of running. Or just click your d-pad one at a time. If you get into an encounter, make sure to remember what step you were on. You can get very lost in here.