Bring a large pot of salted water to a boil. Cook spaghetti until just shy of al dente.
Reserve 1/2 cup pasta water, then drain the pasta.
Warm olive oil and melt butter in a large skillet over medium heat.
Stir in minced garlic and red pepper flakes. Cook gently until fragrant, 1–2 minutes; do not brown.
Pour in chicken broth and simmer until slightly reduced, about 2–3 minutes.
Add heavy cream and simmer until the sauce lightly thickens, 3–4 minutes, stirring often.
Lower the heat. Add the drained pasta and toss to coat.
Remove from heat and sprinkle in Parmesan a handful at a time, tossing until glossy and smooth.
Splash in reserved pasta water as needed to loosen the sauce to your liking.
Season with kosher salt, black pepper, and lemon zest. Finish with parsley and serve hot.