Melt butter in a large skillet over moderately low heat. Add turnips, season with salt and pepper and toss to coat with butter. Cover and cook, shaking skillet occasionally, until turnips are just tender and lightly browned in spots, 10 to 15 minutes.
Add carrots and saute until warmed. Uncover, add shallots and saute 2 minutes to soften the shallots.
While turnips are cooking, bring a large pot of salted water to a boil over high heat. Add green beans and cook until crisp, yet tender, about 5 minutes. Drain well, pat dry, then transfer to skillet with turnips and carrots toss to coat with seasonings.
Taste and add more salt and pepper if necessary. Add parsley, toss again and serve.