Cut carrots and parsnips into 2- x 1/4-inch strips.
Cook carrots and parsnips in boiling water to cover in a large saucepan 8 to 10 minutes or until tender, adding currants, if desired, during last 3 minutes of cooking; drain.
Melt butter in large saucepan over medium heat; stir in mustard, honey, and salt until blended. Stir in carrot mixture, and cook 15 minutes or until thoroughly heated. Sprinkle with chopped parsley.