Set your Instant Pot to Sauté mode and add the olive oil. Once hot, add the onion, carrots, and celery. Cook for 5 minutes, stirring occasionally, until vegetables begin to soften.
Add the minced garlic and cook for 1 minute until fragrant.
Add the zucchini, diced tomatoes with juice, kidney beans, cannellini beans, vegetable broth, Italian seasoning, salt, pepper, and bay leaf. Stir to combine.
Close the lid and set the valve to Sealing. Cook on High Pressure for 5 minutes.
When cooking is complete, do a quick release of the pressure.
Set the Instant Pot back to Sauté mode. Add the pasta and spinach, stirring well. Let it simmer for 8-10 minutes until the pasta is tender and cooked through.
Remove the bay leaf. Taste and adjust seasoning if needed. Serve hot with a generous sprinkle of Parmesan cheese on top.