Bring 3 quarts water to a boil in a large pot and add 1/4 cup salt. Add the pasta and cook until al dente, about 13 minutes.
Meanwhile, heat the broiler. Toss the squash with 1 tablespoon of the olive oil and the garam masala on a foil-lined baking sheet. Broil the squash, 5 to 6 inches from the heat, until browned, 10 to 12 minutes, stirring occasionally.
Reserve 1/2 cup cooking water and drain the pasta. Return the pasta to the pot and toss with the squash, parsley, chile paste, lemon zest and juice and the reserved cooking water.
Divide the pasta among serving bowls and drizzle with the remaining 2 tablespoons olive oil.
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