Bring a large pot of salted water to a boil. Heat the olive oil in a large skillet over medium heat. Add the garlic and cook, stirring constantly, until the edges start turning golden, about 1 minute. Add the tomato paste to the skillet and cook, stirring, 1 minute. Add the red pepper flakes, then add 1 1/2 cups of the boiling water; stir to dissolve the tomato paste. Simmer until reduced by half, about 5 minutes, adding the peas halfway through. Season with salt.
Meanwhile, add the pasta to the boiling water and cook until softened but not cooked through, 2 to 4 minutes. Reserve 1/2 cup cooking water, then drain. Add the pasta to the skillet along with the basil; toss over medium heat, adding more of the reserved cooking water if needed, until the pasta is al dente. Remove from the heat and stir in 1/4 cup parmesan. Top each serving with basil and the remaining parmesan.