Bring an 8-quart stock pot of water to a boil and generously season with salt. Cook pasta according to package directions until al dente. (Stir frequently as fettuccine tends to stick.) If necessary, stir in a drizzle of olive oil. In the last minute of cooking, add broccoli.
Meanwhile, in a large skillet over medium heat melt butter and sauté with garlic for 2 minutes. Add chicken broth, 1 teaspoon salt, and 1/2 teaspoon pepper and bring to a simmer; toss in shrimp and sauté until pink, about 3 minutes. Transfer shrimp to a bowl and set aside.
Stir in tomato sauce and heavy cream; simmer on low for 5 minutes. Toss in pasta until well combined; add back shrimp and broccoli. Stir until warmed through. Garnish with parsley and serve immediately.