Add olive oil to Instant Pot® and select the Sauté function. Set to medium and once hot, add celery, carrots, and onion. Season with salt and black pepper, to taste, and cook for 3 minutes, stirring continually, until the veggies start to soften.
Add mushrooms and dried thyme. Stir to combine and continue cooking for 3-4 minutes, stirring occasionally, until the mushrooms start to develop some color. Turn the Instant Pot® off.
Add wild rice, chicken breast, bay leaves, and broth to Instant Pot®. Add lid and set the pressure valve to “sealing.” Select the “Manual” cooking option on the high setting and set cook time to 30 minutes.
When the cooking time is complete, allow the pressure to release naturally for 10 minutes, and then do a quick release for any remaining pressure. Turn the Instant Pot® off and carefully remove the lid.
Transfer the chicken breast to a platter and shred with two forks.
Return chicken to Instant Pot® and add fresh parsley.
Season with additional salt and black pepper, if desired, and stir to combine before serving. Enjoy!