In this paper, we present a new parametric estimation method for a Lévy moving average process driven by a symmetric α -stable Lévy motion L, α∈ (0, 2 ). More specifically, we consider a parametric family of kernel functions g _θ with θ∈ Θ ⊆ R and propose an asymptotically normal estimator of the pair (α, θ). The estimation idea is based upon the minimal contrast approach, which compares the empirical characteristic function of the Lévy moving average process with its theoretical counterpart. Our work is related to recent papers (Ljungdahl and Podolskij in A minimal contrast estimator for the linear fractional motion. Working Paper, 2018 [14]; Mazur et al. in Estimation of the linear fractional stable motion. Working Paper, 2018 [16]) that are studying parametric estimation of a linear fractional stable motion.