Bring a large pot of salted water to a boil and cook the spaghetti until al dente. Reserve ½ cup of the pasta water before draining.
Meanwhile, heat the olive oil and butter in a large skillet over medium heat.
Add the garlic and cook for about 1 minute until fragrant.
Stir in the lemon zest and red pepper flakes, if using.
Add the crab meat and gently warm it for 2 to 3 minutes, being careful not to break it apart too much.
Pour in the lemon juice and stir gently.
If using, add the heavy cream and simmer for 2 minutes.
Add the cooked spaghetti to the skillet along with the Parmesan cheese.
Toss everything together, adding a little reserved pasta water as needed until the sauce lightly coats the pasta.
Season with salt and black pepper.
Sprinkle with fresh parsley and serve immediately with extra Parmesan and lemon wedges.