Combine broth, water, sliced garlic, red pepper flakes, and ½ teaspoon salt in a large wide skillet or Dutch oven over medium-high heat. Bring to a boil.
Add linguine, fanning strands so they submerge. Reduce heat to medium and cook uncovered, turning with tongs every 2 minutes.
Cook 13–15 minutes until pasta is al dente and 2–3 tablespoons of starchy liquid remain in the pot.
Remove from heat. Scatter in parmesan and toss vigorously with tongs until the cheese melts into a glossy sauce, about 30 seconds.
Add lemon zest, lemon juice, and olive oil. Toss 20 seconds. Taste and adjust salt.
Serve immediately, topped with chopped parsley and extra parmesan if desired.