Parabolic concentrator mirrors are an important component of many solar energy systems, particularly solar mirror collectors. Precision parabolic mirrors are expensive to fabricate and to transport. Here, a new concept for designing and fabricating precision parabolic mirrors is presented. The mirror is formed from a thin flat very flexible metal sheet with a highly reflective surface. Attached to the rear surface of the mirror sheet is a backbone band whose figure is optimized to form the reflective sheet into a precision parabola when its two ends are pulled toward each other. An analytical model to optimize the shape and thickness of the band is presented. The validity of the concept is demonstrated using Finite Element Analysis (FEA) and laboratory experiments. The concept would permit flat mirror elements to be easily fabricated and efficiently packaged and shipped to field sites and assembled into the parabolic trough concentrators with potentially substantial costs reductions compared with the conventional methods.