Bring a large pot of water to a boil and generously season with salt.
Cook pasta according to package directions until al dente. During the last minute of cooking time, add broccoli and cauliflower. Reserve 1 cup pasta water.
Meanwhile, in a separate bowl combine eggs and yolks, cream, olive oil, parmesan and garlic. Mix well with a fork and season with salt and pepper.
Drain pasta and vegetables; add back to pot. Stir in cheese mixture and parsley, then quickly toss together to evenly coat the pasta and prevent the eggs from scrambling.
Add additional pasta water to make a creamy sauce.
Garnish with parsley and pepper. Serve immediately.