The joint parametric estimation of the drift coefficient, the scale coefficient and the jump activity in stochastic differential equations driven by a symmetric stable Lévy process is considered, based on high-frequency observations. Firstly, the LAMN property for the corresponding Euler-type scheme is proved and lower bounds for the estimation risk in this setting are deduced. When the approximation scheme experiment is asymptotically equivalent to the original one, these bounds can be transferred. Secondly, a one-step procedure is proposed which is shown to be fast and asymptotically efficient. The performances in terms of asymptotical variance and computation time on samples of finite size are illustrated with simulations.