Personally, I know very little about parsnips except..
Parsnips lose moisture quickly. With some mud on and without the damage caused by washing, they will keep for two or three weeks in a cool vegetable rack or the bottom of your fridge. Even if they go a bit rubbery this will not detract from the flavour when cooked. From February onwards parsnips are preparing for spring, which means throwing up a seed head from their central core. In preparation, they start re-growing root hairs and sprouting leaf, and the core can start to go tough and woody. So from February it's worth keeping them in your fridge to slow down the clock.
and remember..
If the green tops, or parts of them, are attached, remove them before storing, or they will draw moisture from the roots.