Yes and no actually. I have one, but I'm trying to avoid using it at all costs. That is more or less what I was hoping to see, how much I could accomplish with just the JPG.
I downloaded the photo and took a closer look. Wow. There are actually spots on that bird's head where the blue and green channels are clipped on the low end and the red channel is clipped on the high end -- absolute pure red: R = 255, G = 0, B = 0 and so total loss of detail. I'm guessing you didn't do that in processing the JPEG so your camera's EXPEED processor bares the guilt. You found the perfect photo to illustrate a classic limitation of camera JPEGs. I devote a lot of time to photographing my wife's flowers -- same problem. Rose petals (like feathers) have lots of texture unless the camera obliterates it by clipping color channels. The only solution sans raw file is to adjust the camera controls to damage the photo in ways that can be repaired (forced underexposure and saturation reduction) and then fix that damage as best possible. In the case of this parrot the camera software damaged the photo in a way that can't be repaired. If you restrict the option to camera software processing you're left with two choices that both require you begin with a bad photo. A raw capture is the only way out of that trap.
Joe