Fill a large pot with lightly salted water and bring to a rolling boil. Stir in spaghetti and return to a boil. Cook spaghetti uncovered, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain well in a colander set in the sink.

Meanwhile, heat oil in a large skillet over medium heat. Stir in garlic; cook until softened, 1 minute. Add Swiss chard; cook and stir until chard stems tender. (You can use some of the hot pasta water and cover the pan to help steam the chard.)

Stir hot spaghetti and capers into chard mixture; season with lemon juice, salt, and black pepper. Sprinkle with Parmesan cheese to serve.