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The following should work: for $uv$ with parallel edges connecting them, consider the union of faces not containing vertices, blow it up slightly if necessary to make it disk-shaped and let $\alpha$, $\beta$ be the two $uv$-arcs bounding that disk. Any edge that intersects $\alpha$ and $\beta$ consecutively must intersect all edges between $u$ and $v$. So such an edge can't dip into the edges connecting them. Therefore any such edge which enters the disk must leave the disk via the same arc. You can then shortcut such an edge by following that arc outside the disk. This needs to be done starting with a smallest such arc (in terms of enclosed region inside the disk). Then any new intersections must be with edges it already intersected inside the disk.