Your parsnips have been affected by the leaf skeletoniser moth, Epermenia chaerophyllella. In gardens this insect seems to be more frequently associated with parsnip than any of its other host plants but it can feed on a wide range of plants in the Umbelliferae including carrots.
This insect has two generations during the summer months with larvae usually being found during May to June and again in August to September. The overwintering stage is the adult moth that hides away in sheltered places such as in conifer hedges or other dense shrubs. Where possible the caterpillars can be removed by hand.
The RHS didn't say whether the roots will be affected by the pest.