Wikipedia says [0] it's "due to the weight of the roadway being much greater than that of the cable".

 Yeah, catenary is the shape a cable makes when under its own weight, and with only its own tension holding it up.If you hang a single 1lb weight in the middle of a clothes line, it forms very nearly a perfect V. That's because the 1lb totally dominates the 1/2 oz that the rest of the line weighs.
 Right, but isn't the roadway's weight uniformly distributed along the length of the main cable?Maybe a linear weight distribution favors the parabola, while a curved weight distribution (i.e. the cable's own weight when hanging free) causes the catenary? That makes sense to me, there's more weight per linear meter at the ends than in the center when considering just the cable's weight.
 The reason for this is how the force distributions are applied.A catenary forms when self weight is the driving force. In other words, when the weight per unit length of the cable is the same. For a bridge, the weight of the deck is much alrger than the cable. The weight of the deck is the same per unit horizontal length which is not the same as the unit length of the cable due to the curvature.
 It's uniformly distributed along the length of the roadway. The arc length of the main cable between the suspension cable hanging points is greater near the towers where the slope of the main cable is steep, than closer to the center of the span.

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