Set the Instant Pot to saute function. Let the butter heat inside the pot for 3 minutes. Add the onion to the pot and cook until soft and fragrant, about 3 minutes.
Press the cancel button to stop the sauté setting from continuing to cook.
Add the chicken breasts and the chicken broth to the pot. Deglaze the pot (scrape the bottom with a spoon) and mix well.
Secure lid, close pressure valve and set the timer to 7 minutes on high pressure (use the pressure cook button). It will take the Instant Pot around 9 minutes to build up pressure before it begins cooking.
When cook time is up, do an immediate quick release.
Remove the cover and transfer the chicken breasts to a cutting board. Using two forks, shred the chicken. Add the shredded chicken back to the pot.
Add the cream of mushroom soup, mushrooms, salt and pepper and mix well.
Break the spaghetti noodles in half and add them to the pot, separating them as much as possible with fork tongs and making sure they are completely covered with liquid. Do not stir mixture after this point!
Place the cover on the pot and set timer to 7 minutes on high pressure (pressure cook setting). It will take the IP around 12 minutes to build up pressure before it begins cooking.
When cooking is done, do an immediate quick release.
Remove cover and stir, breaking noodles up if some of them have stuck together. Add the cream cheese and mozzarella and stir until cheeses have melted. Let sit in the pot for 5 minutes.
Top with fresh parsley and serve!