It looks to me like this is achieved by partially desaturating all the colors except the reds. I don't recall the name of the tool in Lightroom, but here's a tool where you can choose to boost or turn down the saturation or lightness of any one of 8 hues. (In Aperture, it's the "Color" filter.)
In Photoshop you could do it with the Hue/Saturation adjustment. From the "Image" menu choose the "Adjustments" sub-menu and then select "Hue/Saturation…". In the "Hue/Saturation" dialog, click on the "Master" menu and choose each color (except red) in turn, and turn down the saturation (but not all the way to -100 or you'll completely remove the color).
Another technique I've used that gives a related result is to convert the image to HSV and apply a curve to just the saturation channel. I usually bring the midpoint down, but leave the end points. That causes things that were not very saturated to lose more saturation, but things that were really saturated to remain saturated. Once you apply the curve, convert back to RGB.
