Place each of the chicken cutlets between 2 pieces of plastic wrap and pound to about 3/8 inch thick. Remove from plastic and season the cutlets with salt and pepper.
Heat the olive oil in a large skillet over high heat. Cook the chicken until its golden brown and cooked through, approximately 2-3 minutes per side. Remove the chicken from the skillet and put it on a serving platter covered loosely with foil to keep it warm.
Turn the heat down to medium, add the garlic and cook until tender, Add the mushrooms and warm them through. Deglaze the pan with the white wine. Using a wooden scraper or spoon, scrape all of the browned bits and pieces from the bottom of the pan. Cook until the wine is almost evaporated.
Add the chicken broth and saffron threads, bring to a simmer and reduce for 8-10 minutes. Add the cream, salt and pepper. Stir to combine and simmer for 1-2 minutes to blend the flavors.
Remove the foil from the chicken and pour the sauce over the chicken. Sprinkle the parsley over the top of the chicken and serve while hot.
