Cook the pasta in a large pot of salted boiling water according to the package instructions until al dente. Reserve about 1 cup of pasta cooking water before draining.
While the pasta is cooking, heat olive oil in a large skillet over medium heat. Add minced garlic and sauté until fragrant but not browned.
Add capers and 1/2 cup pasta water to the skillet and cook for a minute. Add lemon zest to the pan and stir to combine.
Pour in the lemon juice and season with salt, black pepper, and red pepper flakes (if using). Stir to combine.
Add the drained pasta to the skillet. Toss the cooked pasta in the lemon-caper sauce, ensuring it is well coated. If needed, add a bit of the reserved pasta cooking water to loosen the sauce.
Remove the skillet from heat and stir in chopped fresh parsley and grated parmesan cheese. Taste and adjust seasoning if necessary.
Serve the caper lemon pasta immediately, with an extra drizzle of olive oil if desired and grated Parmesan cheese on top.