I use parchment paper from the dollar store when I'm sewing through stuff, it is typically very easy to get off.
How close together is your quilting? Sometimes you can fold and crease straight lines to help. I typically try to rip down one direction and then pull the paper out of the stitching from the other direction if that makes sense.
Among my paper removal tools are popsicle sticks and those manicure type little sticks. For more swirly designs I get those into the paper and zoom around -- but it does create quite a bit of fluff.
The other possibility is to rub along the quilting stitches, but that would be my last resort. Although everything would probably be ok I'd be concerned long term about damage to the quilting thread.
My most recent paper project wasn't quilting but piecing on adding machine paper tape. A light spritz of water and the popsicle stick and the paper was coming off like nothing I've ever seen before, love that paper tape. No little chads stuck in the seam allowane.