Normally for FMQ you just need a hole for the needle, but I'm thinking parchment paper is not as slippery as Teflon or as stiff as Teflon, so maybe the weight of the quilt in combination with manipulating the quilt for FMQ was enough to pull the paper away from the tape so it moved with the quilt. The feed dogs could have helped it along if the stitch length was not set to zero.