Set the Instant Pot to Sauté on high. When hot, add the butter and olive oil and stir until the butter melts.
Add the diced onion, grated carrot, diced celery, minced garlic, and 1 teaspoon of the sea salt. Sauté, stirring occasionally, until the vegetables are softened, about 5 minutes.
Add the cubed chicken, the remaining 1 teaspoon salt, black pepper, Italian dried herbs, and dried parsley. Cook, stirring occasionally, for about 5 minutes until the chicken is lightly browned on the outside.
Pour in the chicken broth and add the Jasmine rice. Stir gently to combine and to distribute the rice evenly.
Place the halved garlic head (cut-side down) on top of the rice, add the bay leaves and one lemon half.
Secure the lid, set the valve to Sealed, and cook on Manual/High pressure for 10 minutes. When done, perform a quick release.
Open the lid, remove and discard the garlic head, bay leaves, and lemon half. Stir the rice and chicken to combine.
Stir in 1/2 cup shredded Parmesan and the chopped parsley until melted and evenly distributed. Serve with additional Parmesan and parsley if desired.