In bug #482 it was also suggested to improve the parabola functionality. I repeat the relevant text here.
[...] By the way there is no reason with bend option to draw a parabola using two cubic curves : any parabola can be drawn by a single quadratic (and so cubic) curve. You can check my answer at TeX.SX.
I think that bend=<point> should be interpreted as "the bend is over the x coordinate of the point". This is backward compatible when there exist a parabola that has a bend at the point. And in all other cases the current parabola operation is drawing a non parabola curve, so there is no need of backward compatibility.
Also, it will be good to add trough=<point> option.