Change parent of a page without unlinking image?

Hello! I'm working on a catalog-style project and have run into a dilemma. It is a large printed brochure that will have 50+ pages using a different parent for left and right pages. Over time, we will need to add pages, but they will be placed in between existing pages as the content is ordered in a specific way—so, pages that were on the left may end up on the right and we'll need to change the parent so the document reads correctly.

The left and right parent pages contain the same image frame but in a different position so the bleed doesn't show in the facing pages. I add an image by dragging it into the frame. However, if I add a page and need to change another page's parent from right to left, this adds a new image frame from the new parent instead of just moving the position of the existing one. This creates a problem as the image bleed appears on the right page. 

I know I can just manually move the images, but am looking to automate this somehow to save time as the document grows. Is it possible to change the parent of a page while keeping the image linked so it adjusts accordingly?