- https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_cur...
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.
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.
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.