The one fact that everyone seems to know about parsnips is that they taste better after they have had a frost through them. The starches turn to sugar, making them sweeter, that's what all the web pages say. But why? With sweetcorn, it's the opposite. You have to run to the pan of boiling water to throw them in before the sugars turn to starch. Vegetables are contrary things.
According to this website (which describes the parsnip as "slightly mucilagenous" - not a great marketing slogan), at low temperatures starches are converted to sugars. This has been known for quite some time, as this textbook from 1867 mentions the change.
This paper from the 1940s looks at how these changes occur in potatoes, but the physiological reason behind the changes from starch to sugar is best explained by this article. Again it looks at potatoes but the reason must be the same - the change of starch to sugar lowers the freezing point of the fluid in the plant cells. This makes the vegetable cells less likely to be damaged by the cold weather (if you want to do an experiment, water with sugar or salt added to it will freeze at a lower temeprature than pure water).
So - great for the plant as it can cope with lower temperatures without its precious storage organ being damaged over winter, and great for us as we get a sweet, tasty veg for our Christmas dinner table.
Just don't try and make parsnip cake.