  We investigate the onset of the Faraday instability in a vertically vibrated
wormlike micelle solution. In this strongly viscoelastic fluid, the critical
acceleration and wavenumber are shown to present oscillations as a function of
driving frequency and fluid height. This effect, unseen neither in simple
fluids nor in previous experiments on polymeric fluids, is interpreted in terms
of standing elastic waves between the disturbed surface and the container
bottom. It is shown that the model of S. Kumar [Phys. Rev. E, {\bf 65}, 026305
(2002)] for a viscoelastic fluid accounts qualitatively for our experimental
observations. Explanations for quantitative discrepancies are proposed, such as
the influence of the nonlinear rheological behaviour of this complex fluid.
