Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add 2 tablespoons oil, rice, and orzo. Saute, stirring and scraping the bottom of the pot constantly, until orzo is brown, about 5 minutes. Pour in 1/4 cup broth and bring to a boil while scraping the browned bits of food off the bottom.
Turn cooker off and add remaining broth, garlic, paprika, and onion powder. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 7 minutes. Allow 10 to 15 minutes for pressure to build.
Meanwhile, heat remaining oil in a medium skillet over medium-high heat. Add mushrooms and saute until lightly browned, about 5 minutes.
Release pressure using the natural-release method according to manufacturer's instructions, for 5 minutes. Carefully release remaining pressure using quick-release method, about 5 minutes. Unlock and remove the lid. Fluff the rice. Add mushrooms and parsley. Taste and adjust salt as needed.