The Leaf Miner is a type of insect that burrows into the leaves of plants like celery and parsnips, feeding between the upper and lower leaf surfaces, thus letting in air and leaving characteristic wiggly whitish / silvery trails. The damage caused is usually not fatal, but it looks unsightly.
|Leaf Miner damage|
I usually only remove any leaves that are badly affected, and then mainly for cosmetic reasons, and I'm not unduly worried by this pest. It doesn't attack the roots of the plants, so they are usually able to survive.
Here's some proof that my parsnips are still OK. Harvested these today. Three nice ones, and one in the "could do better" category...