  The critical behaviour of directed self-avoiding walks is studied on
parabolic-like systems with a free boundary at x=\pm Ct^\alpha. Using a scaling
argument, 1/C is shown to be a marginal variable when
\alpha=\nu_\perp/\nu_\parallel=1/2, i.e., on a parabola. As a consequence the
directed walk may display varying local exponents. Such a behaviour is indeed
observed for restricted walks. This generalizes a result of Cardy showing that
nonuniversal behaviour occurs at corners for isotropic systems.
