Warm the olive oil in a large skillet over high heat. Season the chicken cutlets with salt and pepper. Cook the chicken until golden and cooked through, about 2 to 3 minutes per side. Transfer the chicken to serving plate and tent with foil to keep warm. Turn the heat to medium, add the shallot and the garlic and cook until tender, about 2 minutes.
Deglaze the pan with the white wine. Using a wooden spoon, scrape all the brown bits 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 10 minutes. Add the cream, salt, and pepper. Stir to combine and simmer for 1 minute to blend the flavors. Pour the sauce over the chicken.
Sprinkle with parsley and serve immediately.