The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason, it may be regarded as a basic building block for continuous time models.We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for the whole set of parameters and derive their asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, that is, for the estimation of the successive Blumenthal-Getoor indices and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. Our asymptotic theory is based on new limit theorems for multiscale moving average processes.