SLUGS - They find young plants very attractive. Use slug pellets as required.
CARROT FLY - The tell tale sign are red leaves that wilt in sunny weather. At later stages, the leaves take on a yellow tinge. This pest attacks mainly carrots, but will also go for parsnips and celeriac. Seedlings are killed, and mature roots are riddled with holes, and eventually rot. Attacks are worst in dry soil. No treatment is available once an attack has started. Try to grow carrots well away from tall plants, and sow thinly, ensuring all thinnings are destroyed. If carrot fly are a known problem in your area, lift the early crops no later than mid August, and delay the main crop sowing until late July-early August. Then lift them as soon as they are big enough.
CELERY FLY - Attacks are much less common on celeriac than on celery. All that is usually required is to pick off the occasional blistered leaf. If the problem is severe, lift and destroy the affected plant(s).