Rinse rice with cold water until water runs clear. Using a colander or fine-mesh sieve, drain very well.
Season chicken with salt and pepper to taste. Set aside.
Without the lid on, PRESS the SAUTE mode button on the Instant Pot. When HOT appears on the display, add oil and heat without the Instant Pot lid.
Add onions and cook until softened.
Add mushrooms and cook, stirring occasionally, until lightly browned and liquid dispelled is absorbed.
Add the chicken broth and using a wooden spoon, scrape the bottom of the pot to deglaze.
Add rice, dried parsley, and about 1 teaspoon salt. Stir to distribute.
Arrange chicken breasts in a single layer on top of the rice mixture.
Close and lock the lid, and set the floating valve on the SEALING. Pressure cook on HIGH for 5 minutes.
When finished, do a 10-minute NATURAL RELEASE. Then turn the steam floating valve on the lid to the VENTING position to release the remaining pressure manually. Carefully open lid.
Using two forks, shred the chicken and gently fluff the rice to distribute. Serve hot.