We like to eat any peels that are clean, tender, and tasty enough, believing that they contain good nutrients and fiber. Some vegetables, like celery root, have tough outer peels, which we always remove. Parsnips we assess on a case-by-case basis; the best flavor is actually right below the skin, so we try to avoid removing too much. As for carrots, we usually just scrub them well and leave the peel on.
Our decision to peel also depends on where we got the produce and whether it's conventionally or organically grown. Foods like cucumbers, bell peppers, eggplants, and potatoes might also be coated with wax and we'd rather not munch on that!
Related: Gallery: How To Peel Things
(Image: Flickr member Connors75, used by permission)