  Heteroclinic orbits for one-parameter families of nonautonomous vectorfields
appear in a very natural way in many physical applications. Inspired by some
recent bifurcation results for homoclinic trajectories of nonautonomous
vectorfield, we define a new $\mathbf Z_2$-index and we construct a index
theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index
theorem, by showing that, under some standard transversality assumptions, the
$\mathbf Z_2$-index is equal to the parity, a homotopy invariant for paths of
Fredholm operators of index 0. As a direct consequence of the index theory
developed in this paper, we get a new bifurcation result for heteroclinic
orbits.
