Heat oil in a large pot set over medium-low heat. Add onion and garlic and cook, stirring often, until onion and garlic are softened, about 8 minutes.
Add beans, broth, marinara, and Parmesan rind. Increase heat to a simmer and cook, stirring often, for 15 minutes.
Meanwhile, bring 2 quarts of water to a boil in a large pot. Season with 2 teaspoons salt. Add kale and cook until tender, about 3 minutes. Use tongs to transfer kale to the soup, reserving blanching water.
Return water to a boil, add pasta, and cook following package directions. Drain pasta and portion among serving bowls.
Stir vinegar into the soup and season to taste with salt and pepper then ladle over pasta. Garnish with grated Parmesan and crushed red pepper, if using.