Preheat the oven to 425°F (220°C) and line a large sheet pan with parchment paper.
Cut the carrots into even sticks of the same thickness so they roast at the same rate, and trim the woody ends from the asparagus.
In a large bowl, toss the carrots and asparagus with the melted butter, olive oil, garlic, thyme, salt and black pepper until every piece is evenly coated.
Spread the vegetables in a single layer on the prepared sheet pan, making sure the pieces do not touch, so they roast and caramelize instead of steaming.
Roast for 20 to 25 minutes, turning the vegetables halfway through, until the carrots are tender with caramelized edges and the asparagus is tender-crisp. If using the Parmesan, sprinkle it over the vegetables for the last 5 minutes of roasting.
Transfer to a serving dish and finish with the chopped parsley and a squeeze of fresh lemon juice. Serve immediately.