Sauté the aromatics: Add olive oil to the bottom of the Instant Pot and turn on the Sauté setting. Once hot, add the diced onion and cook for about 5 minutes, until lightly softened and golden.
Add vegetables and seasoning: Stir in the chopped carrots, celery, zucchini, Italian seasoning, salt, and pepper. Cook for an additional 3–4 minutes, stirring occasionally to prevent sticking.
Cancel sauté mode: Press the Cancel/Keep Warm button to stop the sauté function.
Add remaining ingredients: Pour in the vegetable broth, crushed tomatoes (with juice), tomato paste, shredded kale, bay leaves, and ditalini pasta. Stir to combine. Do not add the white beans yet.
Pressure cook: Secure the lid on the Instant Pot and set the valve to Sealing. Select the Manual or Pressure Cook setting and cook on High Pressure for 7 minutes.
Natural release: Allow the pressure to release naturally for 10 minutes, then carefully perform a quick release to release any remaining steam. Remove the lid.
Add the beans: Stir in the drained white beans and adjust seasoning with more salt and pepper to taste.
Serve: Ladle the soup into bowls and top with fresh chopped parsley and grated Parmesan cheese, if desired.