Wouldn't thicker sidewalls weigh more than thinner sidewalls since they are so much bigger?
One source of tire rolling resistance has to do with the details of the movement of the tire outside surface, within the contact patch, as the tire rolls. Previous posts have correctly noted that the tire contact area is unchanged by the width of the tire. i.e., for a given weight on the tire & tire pressure, a wider tire will have a contact patch that is wider, but shorter and with the same area as the narrower tire. By shorter, I mean shorter along the direction of travel. This change in shape is important to remember a bit later.
As the tire rolls, the edges of the tire contact patch are pulled forward while the center tends to be pulled aft and most of the tire patch tends to scrub slightly because the diameter of the tire is slightly larger in the center then at the edges. This is one reason why a narrow tire contributes to better fuel mileage. With this understanding, one can see how higher tire pressures can contribute to better fuel mileage thru this effect: With higher tire pressure, the area of the contact patch is reduced, particularly the length of the patch, which minimizes the scrubbing.
Of course, other factors of the tire also affect fuel mileage. A narrow tire can have slightly lower aerodynamic drag. However, this is likely to be a very small effect given the airflow environment that the tire operates in. A more productive way to reduce aerodynamic drag associated with the tires / wheels would be to block the holes that allow air passage thru the wheel. But, this would adversely affect brake cooling.
Since the tire is a complex visco-elastic structure, the rolling deflection of the tire will consume energy. Higher tire pressure reduces this source of energy loss by reducing the magnitude of the tire deflection with rolling.