Add linguine to a large pot of salted boiling water and let cook until al dente, about 7 minutes. Drain well, reserving about 1 cup of the pasta water.
While pasta is cooking, add 2 Tbsp butter and olive oil to a large skillet over medium heat. Add shrimp to pan in a single layer (you can cook it in two batches if necessary) and sprinkle with ½ tsp salt, and ½ tsp pepper. When shrimp start turning pink (about 2 minutes), flip and cook for another minute or so on the other side, until opaque. Remove shrimp to a plate or bowl and set aside.
Add remaining 2 Tbsp butter to skillet and once melted, add garlic. Cook for 1-2 minutes, until it's just turning golden
Add heavy cream, lemon zest, lemon juice, and black pepper to skillet and gently simmer, stirring occasionally, for about 5 minutes, until sauce is thickened.
Remove from heat and add pasta to skillet. Toss to coat in sauce. Add ½ cup of reserved pasta water and continue to toss (add up to an additional ½ cup of pasta water if needed for consistency). Add shrimp and parsley to skillet and toss to combine.
Serve pasta in bowls or on plates with additional lemon zest, black pepper, and parsley.