Season chicken breasts with 1/2 teaspoon salt and 1/4 teaspoon pepper. Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Heat 1 tablespoon butter and 1 tablespoon olive oil; cook chicken breasts on both sides until golden brown, about 5 minutes. Remove chicken from the pot. Melt remaining 2 tablespoons butter and 1 tablespoon olive oil; add garlic and shallots; cook until softened, about 3 minutes. Stir in mushrooms and cook until tender, about 3 minutes. Pour chicken broth into the pot and bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon for deglazing, about 3 minutes. Return chicken to the pot and cover with Marsala wine. Close and lock the lid. Select Poultry function; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, about 10 minutes. Release remaining pressure using the quick-release method, about 5 minutes. Unlock and remove the lid. Select Saute function. Dissolve cornstarch in water and stir into the pot with heavy whipping cream. Season with remaining teaspoon salt and 1/4 teaspoon pepper; cook until sauce has slightly thickened, about 5 minutes. Serve and sprinkle with fresh parsley.
