Preheat your oven to 425°F. Line your sheet pan with parchment paper. Place the pan in the oven while it preheats — about 5 minutes.
In a large bowl, combine the halved potatoes and carrot chunks. Add 2 tablespoons of olive oil, the rosemary, thyme, and 1/2 teaspoon of salt. Toss well to coat everything evenly. (Photo tip: You should see each piece glistening with oil – not pooled at the bottom of the bowl. If there's excess oil in the bowl, you've used too much.)
Carefully remove the hot sheet pan from the oven (watch your fingers!). Spread the potatoes and carrots in an even layer, making sure they're not crowding each other. Add the smashed garlic cloves to the pan. Roast for 15 minutes.
In the same bowl you used for the potatoes (no need to wash it), toss the zucchini half-moons with the remaining 1 tablespoon of olive oil and 1/2 teaspoon of salt. This separate tossing is important — zucchini needs less oil than the root vegetables, and doing it separately means everything gets exactly the right amount.
After 15 minutes, pull the pan out of the oven. Give the potatoes and carrots a quick stir. Add the zucchini in an even layer, spreading them out so they have room. Don't pile them on top of the potatoes — give them their own space to brown.
Return the pan to the oven and roast for another 12-15 minutes. The zucchini should be tender and golden at the edges, and the potatoes should be crispy on the outside and easily pierced with a fork. (Photo tip: Look for the deep golden-brown edges on the potatoes and the slight browning on the zucchini — that's where the flavor lives.)
For that final shatteringly crispy edge, switch the oven to broil for 1-2 minutes. Watch it closely — it goes from perfect to burnt very quickly.
Remove from the oven. Sprinkle with fresh parsley and a final pinch of salt. Serve immediately while everything is hot and crispy.