Hi Leif, last spring I had a similar strange thing happen in my carrot patch which I had completely netted with micromesh over large hoops and tied off the ends. My neighbour found a magpie in there, and it had caused similar destruction to the plants that you describe without damaging the net structure. How it got in (and why it wanted to?!) is beyond me! They are big strong birds, though. So I wonder if this may be what happened with your parsnips too?
Good luck with recovering your seedlings!