Combine chicken broth, chicken, and rice in a multi-functional pressure cooker (such as Instant Pot). Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 4 minutes. Allow 10 to 15 minutes for pressure to build.
Meanwhile, whisk eggs and Parmigiano-Reggiano cheese together in a small bowl. Season with black pepper. Set aside.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Change the pressure cooker setting to Saute. Pour egg mixture into the broth in a slow steady stream while whisking. Cook for 1 minute. Serve immediately and garnish with parsley.