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.