This is my favorite recipe for parsnip - roasted with horseradish
. If you look at that site, there are several other parsnip recipes, too. You can also make parsnip fries with them (baked) - never tried them fried.
Radishes - red radishes can be cooked like baby red potatoes. They are not starchy, so will not absorb flavorings, but when they're cooked they get tender and lose their bite so flavor is very similar. Great for things like faux-tato salad where the potato is not contributing anything other than volume. Daikon radish can also be cooked - I've made faux-tato pancakes with it, and I've also baked it into casseroles (chunked) instead of potatoes.
Cauliflower is the other typical potato replacer - does great steamed and mashed. I prefer to only cook it until soft (and not mushy), and then mix it with cream cheese for the creaminess that is missing from the potato starch - add some garlic and blue cheese for flavor. 1 head of cauliflower really cooks down a lot once mashed - if it's more than just the two of us I always cook at least 2 heads (for 4 adults).
Are tomatillos nightshades? That's the base of most green salsas I'm familiar with. One of my favorites (that I've had but never found a recipe for) uses tomatillos, avacadoes and hard cooked eggs. It's the spiciness of salsas that'll be difficult without nightshades. But there are dozens of types of salsas that can be made without nightshades.
Personally I don't care for rutabagas or turnips - hopefully someone else will have some thoughts on their use.
Oh, another potato replacer is plantain - the riper the plantains the sweeter - the greener they are, the more starchy. I make mofongo - I just sautee up some bacon and onions, toss in chunked peeled plantain, add some stock, cover and let it simmer until they're soft. Mash the plantain, add a little salt and it makes a great mashed potato replacer. They can also be fried - either as chips (sliced paper thin), or as a veggie (sliced thicker). The riper they are the sweeter and softer they are, so they only need to be cooked until brown. If they're green they usually need to be cooked twice to soften up. These are great served with sour cream.
And of course there's squash. Shredded summer squash works decently well as hashbrowns/potato pancakes or other dishes that call for shredded potatoes (squeeze the water out first). Winter squash cooked and mashed is a less starchy alternative to sweet potatoes - they can be used fairly interchangeably in most recipes.
For curry, you'll have to blend your own to eliminate the nightshade.