Boil a large pot of salted water and cook the whole wheat pasta according to package instructions until al dente. Reserve 1/2 cup of pasta water before draining.
Season the cubed chicken with salt, black pepper, and dried oregano.
Heat olive oil in a large skillet over medium-high heat. Add the chicken and cook until browned and internal temperature reaches 165°F (74°C), about 6-8 minutes.
Remove the chicken from the skillet and set aside on a plate.
Reduce the skillet heat to medium. Add the minced garlic and red pepper flakes, sautéing for 1 minute until fragrant.
Pour in the chicken broth to deglaze the pan, scraping up any browned bits from the bottom. Simmer for 2 minutes.
Reduce heat to low. Whisk in the Greek yogurt and Parmesan cheese until the sauce is smooth and the cheese has melted. Do not boil, as the yogurt may curdle.
Add the cooked pasta, chicken, and baby spinach to the skillet.
Toss all ingredients together until the spinach is wilted and the pasta is evenly coated. Add reserved pasta water 1 tablespoon at a time if the sauce needs thinning.
Garnish with chopped fresh parsley and serve immediately.