Bring a large pot of salted water to a boil.
Melt butter in a large skillet over medium heat, then add garlic, olive oil, lemon zest, lemon juice, and salt. Turn off heat.
Cook pasta in boiling water until 1 minute under al dente. Reserve 1 cup pasta water before draining.
Turn skillet back to medium heat and add heavy cream and 3/4 cup pasta water to lemon butter mixture.
Simmer for 3 minutes until sauce thickens enough to coat the back of a spoon.
Add drained pasta and Parmigiano-Reggiano to skillet and toss vigorously for 1 minute until sauce is glossy and coats pasta evenly.
Add more pasta water if sauce is too thick.
Garnish with parsley and black pepper.