Heat the olive oil in a large wide pot over medium-high heat. Add the chard stems, season with salt and cook, stirring occasionally, until tender, about 5 minutes. Add the garlic and red pepper flakes and cook until the garlic is softened, about 30 seconds. Stir in the tomato paste and cook until slightly darker in color, about 1 minute. Carefully add the vodka and cook 30 seconds to burn off the alcohol but not all the liquid.
Add 4 cups water, the crushed tomatoes, 1/2 teaspoon salt and the spaghetti to the pot and stir. Bring to a boil, then reduce the heat to a simmer; stir in the chard leaves. Cook, stirring occasionally, until the pasta is tender and the sauce is thickened, 10 to 12 minutes.
Stir in the heavy cream and parmesan. Cook, tossing, until combined. Remove from the heat and stir in the cherry tomatoes. Let stand a few minutes to let the sauce thicken; season with salt. Divide among bowls and top with more parmesan.