Who is eating these tunnels in my carrots? They all called the carrot weevil. The adult in a tiny brown bug with a snout laying eggs on the crown of the carrot. The little white worms burrow down on the taproot leaving tunnels along the root.
If you have them this year, you will have them next year. The best control is prevention by digging up the garden in the fall and removing the debris from the garden. Crop rotation in the garden is the best way to prevent many problems. They are worse in wetter years than dry years.
You can use a drench of Neem oil (mixed with equal parts of water in a sprinkling can) and saturate the ground where the carrots are growing. If you see the weevils before they lay eggs, you can spray with Malathion.
These weevils attack parsnips, but to a much lesser extent.