The reason to peel potatoes is not phytic acid (the anti-nutrient you are talking about) in the skin, but rather the solanine (potato toxin) in the skin (and who knows what types of fungi and molds make up the random array of surface defects on a potato). Phytic acid is not so great if you are mineral deficient and eating phytic-acid rich foods as staples but, otherwise, there's nothing wrong with it (in fact, it might be quite heathy).

I don't know what lurks in the skins of carrots and parsnips but I do peel them anyway simply because they are usually covered in dirt (and maybe chemicals).

Wash them with a veggie brush under flowing water and you are good. That is what I do with carrots too.

Potatoes get peeled because of the risk of green nightshade compound pockets. I have been told everything else has lots of nutrition in the skin.

Here the winter parsnips are so woody that i always peel them.