The skin of most fruits and vegetables are full of nutrition and fibre that you miss out on when you peel them, but when someone reaches for a peeler after picking up a carrot, beet, or parsnip, no one bats an eye. Put the peeler down — you don't have to use it.
Image from wwworks.
If you're worried about pesticides (and you probably shouldn't be), peeling is a quick way to remove them, but only from the surface. If you're not concerned about that, then the way you plan to cook your vegetables should determine whether to leave the skin on or not. Rochelle Bilow, writing for Bon Appetit, explains:
Some methods of cooking fare better than others for skin-on vegetables than others. Here's when it's ok to leave the skin on: Roasting, mashing, and, depending on the variety, grating or chopping raw. Here's when you should probably use the paring knife or peeler: Steaming (the skin can be tough if steamed), puréeing — for example, this potato recipe — and raw preparations of veggies with extra-thick skin.
Some vegetables, like beets, do have bitter tasting skins. You can tone this down by dressing them with acid and sweetness. While there are many vegetables you don't need to peel, ones with very fibrous or tough skin should always be peeled, such as kabocha squash or celeriac.