Set the instant pot to the sauté setting and add the oil. Sauté the onion, celery and carrots for 5-7 minutes or until softened.
Add the garlic and sauté for an additional minute.
Add the tomato paste and stir to combine. Continue to cook on the sauté setting for 2-3 minutes while stirring.
Cut the rind off of the slice of Parmesan and set the cheese portion aside. Cut the rind into 2-3 smaller pieces (I find this tactic maximizes the cheesy flavor as it infuses into the broth) and add it to the soup. Grate or shave the cheese so that it's ready to top each bowl of soup once it's done cooking.
Add all remaining ingredients to the instant pot except for the optional pasta.
Set the instant pot to the pressure cooking setting for 20 minutes.
Once the pressure cooking process is complete, vent the instant pot using the quick release. Remove the lid and remove the stems from the rosemary and any large pieces of Parmesan rind that are left.
Serve each bowl of soup with 1/3 cup cooked pasta (optional) about 1-2 tablespoons of grated or shaved Parmesan cheese.