In a large skillet over medium heat, heat olive oil. Add chicken and season generously with salt and pepper. Cook until no longer pink, 5 minutes per side.
Meanwhile, bring a large pot of salted water to a boil. Cook spaghetti according to package directions until al dente. Drain, reserving 1 cup pasta water, and set aside.
Transfer chicken to a plate and add butter to skillet. Let melt, then add garlic.
Cook, stirring constantly, until light brown, then add lemon juice and zest. Stir in crushed red pepper flakes. Pour in 1/4 cup reserved water - scraping all the bits on the bottom of the pan.
Cut chicken into bite size pieces and return to skillet and gently.
Add spaghetti and heavy cream - toss until combined.
Top with with parmesan cheese.
Serve immediately with a lemon wedge and a sprinkle of parsley.