Preheat your oven to 400°F (200°C) and lightly coat a large rimmed baking dish or sheet pan with nonstick spray or a small amount of oil.
Place the halved baby potatoes in a mixing bowl. Add 2 tablespoons of olive oil, half the minced garlic, the Italian seasoning, paprika, salt, and pepper. Toss until evenly coated, then arrange them on one side of the prepared pan in a single layer.
Drizzle the remaining tablespoon of olive oil over the chicken breasts. Rub each piece with the leftover garlic, a touch of salt and pepper, and sprinkle with half of the Parmesan cheese. Lay the seasoned chicken pieces on the opposite side of the pan from the potatoes.
Place the pan in the oven and roast for 30 to 35 minutes. Turn the potatoes halfway through cooking. The dish is ready when the chicken reaches an internal temperature of 165°F (74°C) and the potatoes are fork-tender with a golden finish.
Remove the pan from the oven. Brush the chicken and potatoes with melted butter and scatter the remaining Parmesan cheese evenly across the top.
Finish the dish by garnishing with freshly chopped parsley. Serve hot.