Preheat oven to 400°. Toss together first 5 ingredients and 1 Tbsp. olive oil in a small bowl.
Cut carrots and parsnips lengthwise into long, thin strips.
Whisk together brown sugar, balsamic vinegar, and remaining 3 Tbsp. olive oil in a large bowl. Toss with carrots and parsnips, and place on a lightly greased 15- x 10-inch jelly-roll pan. Sprinkle with desired amount of salt and freshly ground pepper.
Bake at 400° for 40 to 45 minutes or until vegetables are tender and browned, stirring every 15 minutes. Transfer to a serving platter, and gently toss with feta cheese mixture.