Bring a large pot of salted water to a boil and cook the angel hair pasta for 3 to 4 minutes until al dente, then drain and reserve 1/2 cup pasta water.
While the pasta cooks, heat olive oil in a large skillet over medium-high heat for 1 minute, then add garlic and cook for 30 seconds until fragrant.
Add cherry tomatoes and cook for 3 minutes, stirring occasionally, until they start to soften and release juice.
Stir in baby spinach and cook for 1 minute until just wilted.
Add the drained pasta, lemon juice, lemon zest, red pepper flakes, and 1/4 cup reserved pasta water, tossing for 1 minute until well combined.
Remove from heat, sprinkle with Parmesan cheese, and serve immediately.