The scattering of high-frequency sound waves by two-dimensional curved boundaries has received much attention over the past few decades, with particular interest in the effects of tangential ray incidence. In the event that the radius of curvature is not small, an analysis near the point of tangency gives rise to the Fock–Leontovič equation for the local field amplitude which, in turn, matches the creeping field of Keller's geometrical theory of diffraction. If the radius of curvature is sufficiently small, however, then this analysis is not valid and it is necessary to solve the full Helmholtz equation in the presence of a parabolic boundary. Under these conditions, which are canonical for diffraction by a sufficiently slender body, results are presented for the case of a plane wave impinging upon an acoustically hard parabolic cylinder. This diffraction process engenders a creeping field at one tip of the slender body, which then propagates around the body to the other tip. Here its energy is partially reflected, partially transmitted and partially radiated out in a detached field. A full description of this is given, along with a discussion of the ‘blunt’ limit in which we show that not only do we get the traditional creeping field of Keller's geometrical theory of diffraction, but also an exponentially small backward-propagating creeping field not predicted by traditional ray methods.