In a firepit, build a wood fire about as wide as the cooking grate. Let it burn to low flames and embers (1 to 1 1/2 hours); fire is ready when you can hold your hand 5 in. above grate only 3 to 5 seconds. Or use a charcoal fire that’s about this hot.
Meanwhile, put potatoes in a medium saucepan and add 1 tsp. salt and water to cover. Cover pan and bring to a boil on a camp stove. Reduce heat and simmer until barely tender, 3 minutes; drain.
Lay a 12-in. square of heavy-duty foil flat on a work surface. Drizzle center with a little olive oil. Spoon about 1 cup potatoes and 1/6 of the onion and carrots on top. Season generously with salt and pepper. Set aside 1 tbsp. parsley for serving. Sprinkle vegetables with about 1/6 of remaining parsley, rosemary, and thyme. Drizzle with 1 tbsp. olive oil.
Bring 2 opposite sides of foil together across the middle and fold to make a seam about 1 in. wide. Fold seam on itself again, then fold it flat on packet. Fold ends over themselves, tucking them into a compact packet. Repeat to make remaining packets.
Set bundles on grill and cook, turning with tongs every 3 to 5 minutes, until vegetables are tender and browned, about 15 minutes (open carefully to check).
Protecting hands, snip off ends of packets with scissors (or use a knife) and pour food onto plates. Sprinkle with about half the parmesan and the remaining parsley, and tear basil leaves on top. Serve with more cheese on the side.