  The well known theorems of Khintchine and Jarn\'ik in metric Diophantine
approximation provide comprehensive description of the measure theoretic
properties of real numbers approximable by rational numbers with a given error.
Various generalisations of these fundamental results have been obtained for
other settings, in particular, for curves and more generally manifolds. In this
paper we develop the theory for planar curves by completing the theory in the
case of parabola. This represents the first comprehensive study of its kind in
the theory of Diophantine approximation on manifolds.
