Season the chicken breasts by combining the dried oregano, salt, and black pepper in a mixing bowl, then rub this mixture evenly over both sides of each chicken breast.
Set the Instant Pot to the 'Sauté' setting, add the olive oil, and once hot, add the minced garlic. Sauté for about 30 seconds until fragrant, being careful not to burn it.
Place the seasoned chicken breasts into the Instant Pot in a single layer and sauté for about 2-3 minutes on each side until lightly browned, potentially in batches.
Pour in the chicken broth, then squeeze in the lemon juice and add the lemon zest. Scrape up any browned bits from the bottom of the pot to prevent a burn warning.
Secure the lid on the Instant Pot and set the valve to the sealing position. Cook on high pressure for 6 minutes.
Once the cooking time is up, allow the Instant Pot to naturally release pressure for 5 minutes, then carefully switch the valve to the venting position to release any remaining pressure.
Open the lid and use tongs to remove the chicken breasts. Allow them to rest for a few minutes before slicing.
Serve the chicken on a platter, drizzling the lemon-garlic sauce from the pot over the top, and garnish with fresh parsley.