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I let a huge parsnip go to seed to collect the seeds for the spring (collected a week or three ago), but many have fallen to the ground and immediately germinated into strong little plants.

What about leaving them to sit there over the winter and grow away in the spring instead of putting up with difficult germination in the spring? Has anyone tried it?

I haven't personally but chap next door did..and we had parsnips 'flowering' all over ...they did  not grow  'proper' roots..
The winter cold will trigger the biennial growing patern...so autumn is the year 1 and next spring/summer the flowering and setting seeds..job done for them..now if they would have seeded earlier on and they would have had longer growing season you might have had some roots to eat..but as for now it's a bit late..sorry..

Thanks - I suspected what the answer would be, but it is very useful to have confirmation from experience. Onto the compost heap, then.

Remember to store the seed you have collected really well, as you must have very viable seed if they self seeded, as we all know parsnips can be difficult to germinate.

If it was me I'd be tempted to put a plaggy cloche over them for the winter and see what happens.  It's not until mid-November normally, in my experience down here, that carrot and parsnip stop growing and with a cloche you might go on for a bit longer.  

After all, you can overwinter carrots from an autumn sowing under some protection though it doesn't seem the best way to grow them.