Heat the olive oil in the Instant Pot using the Saute function. Add the onion, carrots and celery and saute until softened, stirring often, about 5 minutes.
Add the garlic, dried thyme, cumin, smoked paprika, salt and pepper. Cook, stirring, for 30 seconds and then press Cancel to turn off the Saute function.
Pour in the broth and scrape up any bits that are stuck to the bottom of the pot. Add the crushed tomatoes and lentils and stir.
Close the Instant Pot lid and turn the steam release valve to the sealing position. Set the cook time to 8 minutes at high pressure using the Pressure Cook or Manual function. The Instant Pot will take about 25 minutes to reach pressure and then the cook time will begin counting down.
When the cook time ends, allow the pressure to release naturally for 10 minutes by leaving the Instant Pot alone. Then carefully turn the steam release valve to the venting position (I use the handle of a long spoon to do this).
Wait for all of the steam to escape and the pin to drop down and then carefully open the Instant Pot lid. Stir the chopped spinach into the soup. Serve with grated Parmesan, if desired.