Slice the cauliflower lengthwise, about 1/8-inch thick. Save the little cauliflower "crumbs". Thinly slice the parsnips.
Heat 2 tablespoons of olive oil in a large skillet over medium heat. When hot, but not smoking, add the cauliflower slices and parsnip slices, spreading out into a single layer, if possible. Do not stir, but let the vegetables cook, undisturbed, until nicely browned on the bottom, about 3 minutes. Add the salt, reserved cauliflower crumbs, garlic, and red pepper flakes, and give the vegetables a gentle stir. Continue cooking, only stirring once or twice, until the vegetables are cooked through and nicely browned, about 5 minutes more. Serve hot.