Consider an algebraic ordinary differential equation (AODE), i.e. a polynomial relation between the unknown function and its derivatives. This polynomial defines an algebraic hypersurface. By considering rational parametrizations of this hypersurface, we can decide the rational solvability of the given AODE, and in fact compute the general rational solution. This method depends crucially on curve and surface parametrization and the determination of rational invariant algebraic curves. Transforming the ambient space by some group of transformations, we get a classification of AODEs, such that equivalent equations share the property of rational solvability. In particular we discuss affine and birational transformation groups.