Heat oil in a large, deep skillet over medium. Once hot, add garlic, stirring occasionally, until lightly browned around the edges, about 4 minutes. Add zucchini and squash; increase heat to medium-high; season with the salt. Cook for 12-15 minutes, stirring occasionally, until the squash is jammy and soft. Sprinkle with chili powder and mix.
Meanwhile, cook pasta in a large pot of boiling water, stirring occasionally, until very al dente.
Transfer pasta to skillet with squash using a slotted spoon and add ½ cup pasta cooking liquid. Cook pasta, add Parmesan and more pasta cooking liquid as needed, until sauce coats pasta and pasta is al dente. Toss in lemon juice and most of the basil.