My experience with parsnips has been more towards avoiding the biggest ones because they go rough and woody inside which isn't nice. Peel wise I've never noticed an issue with carrots, and I come from a household where mum was one of those food fad mums. Nothing was peeled if the peel was edible.
I only peeled the potatoes on the recipe I made recently (along with the almonds) because potato peel in a cake didn't appeal to me. Usually even mashed potato would not be peeled in this house and both of us ignore any peel instructions in recipes. I have several that say to peel aubergine! No way. Guess I'm just used to making the most of as much add possible, plus a lot of these veg, the peel is where half or so of the vitamins and minerals are.