Preheat the oven to 425°F and position racks in the upper and lower thirds.
Line two large sheet pans with parchment for easier cleanup.
Cut vegetables into similar 1-inch pieces and pat them dry to remove excess moisture.
Whisk olive oil, garlic, salt, pepper, Italian seasoning, smoked paprika, and red pepper flakes in a large bowl.
Toss all vegetables in the seasoned oil until evenly coated.
Spread vegetables on the pans in a single layer, leaving space for browning.
Roast for 15 minutes. Stir, rotate the pans, and roast 10–12 minutes more until tender with crisp edges.
Finish with lemon juice, then sprinkle parsley and Parmesan. Taste and adjust salt if needed.