What is the shape of a rope hanging from two ends?

This is a classic problem in the calculus of variations, and the shape is not, in fact, a parabola. The curve is called a catenary and its basic equation is $$ y=a \cosh(x/a). $$ For more details, see the catenary page at Wikipedia.

It can be shown using the calculus of variations that this is indeed a catenary. Written as

$$ y = a \, \cosh \left ({x \over a} \right ) = {a \over 2} \, \left (e^{x/a} + e^{-x/a} \right )\, $$

A slightly more interesting problem arises in stretching soap film between two concentric circular wires. Due to the axial symmetry of this problem the solution is the catenoid. This is the surface of revolution obtained by rotating the catenary.

I would highly recommend Gelfand and Fomin "Calculus of variations" (Dover Publications) for further reading on such problems.