Set the Instant Pot to sauté mode. Add the onion, carrots, celery, and garlic. Cook for 3 minutes, stirring often, until the vegetables begin to soften.
Add the chicken breasts, wild rice, chicken broth, thyme, bay leaf, salt, and pepper. Stir once to combine.
Secure the lid, set the valve to sealing, and cook on high pressure for 25 minutes.
Let the pressure release naturally for 10 minutes, then quick release any remaining pressure.
Remove the chicken breasts to a cutting board and shred them with two forks. Discard the bay leaf.
Return the shredded chicken to the pot and stir everything together. Ladle into bowls and top with fresh parsley.