Don't know about celery, though I would think not. Parsnips will be fine left in the ground, in fact I think they always taste better when they have been frosted. If you must lift store somewhere cool and dry in a box of sand.

We freeze Parsnips. Saves having to dig them up when it is cold!. Parboil the roots and then freeze. The act of freezing them by the way has the same effect as them freezing outside.
Celery can be frozen but it is no use afterwards other than as an ingredient in soups etc..

iv.e cooked drained and froze celery and used it later as a veg. fried in butter . yum