A few pantry staples—apple cider vinegar, dry mustard, and paprika—turn this traditional side into something spectacular. Be careful not to over-reduce the glaze; it should have a thin, syrupy texture.
Place carrots in a large saucepan with water to cover, and bring to a boil over high. Reduce heat to low, and simmer until tender, 3 to 5 minutes. Drain and set aside.
Combine brown sugar, apple cider vinegar, butter, salt, dry mustard, and paprika in a large nonstick skillet over low; cook, stirring often, until butter melts. Increase heat to medium-high, and bring to a boil. Reduce heat to medium; add carrots. Cook, stirring constantly, until carrots are glazed and sauce is syrupy, 3 to 4 minutes. Remove from heat. Stir in walnuts; sprinkle with chopped parsley, and toss to combine. Serve immediately.