We had a couple of derelict plots on our allotment that were always knee-deep in self-sown parsnips. I even sowed a few seeds myself (they were pretty nice parsnips, actually) - I guess their evolutionary strategy is just to produce absolute manure-loads of seeds, and some will be OK.
One of the disadvantages of being fully allocated is you lose the weird and wonderful self-sown plots - though that depends on what they have on them! Blackberries.. great! Hedge bindweed.... Grrr!