Press "sauté" on the Instant Pot and then add the butter. Let the butter melt and then add the chopped onion, garlic and carrot.
Sauté the garlic, carrots and onions for 3-5 minutes. They don't need to be soft.
Add the brown rice, Italian seasoning, salt, and pepper and continue to sauté for another 3 minutes. Push "cancel" on the Instant Pot to turn it off.
Add the chicken broth, mix well, and scrape the bottom and edges of the insert pot. Make sure all of the rice and veggies are in the liquid and are not stuck to the side or bottom of the insert.
Add the chicken on top of the rice mixture. Do not mix. Just leave the chicken on top.
Close the Instant Pot lid and set the valve to "sealing". Program for manual-high and 22 minutes.
Allow the pressure to release naturally for at least 10 minutes after the cook time has ended, then release the rest of the pressure by carefully moving the valve to "venting".
Remove the lid and remove the chicken to a cutting board. Add the frozen peas and the chopped mushrooms to the rice mixture in the Instant Pot and stir well.
Put the lid back on the Instant Pot and move the valve to sealing. Do not turn the Instant Pot on. Leave sealed for 5 minutes and then release the pressure that has built up by moving the valve to venting.
Chop the chicken or shred with 2 forks. Remove the lid of the Instant Pot and add the chicken back to the pot and mix well.
Serve in bowls with fresh parsley and Parmesan.