In this paper we present some limit theorems for power variation of a class of stationary increments Lévy driven moving averages in the setting of critical regimes. In an earlier work the authors derived first and second order asymptotic results for kth order increments of stationary increments Lévy driven moving averages. The limit theory heavily depends on the interplay between the given order of the increments, the considered power p > 0), the Blumenthal–Getoor index β ε (0,2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α .In this work we study the critical cases α = K-1/p and α = k-1/β with p ≠ β ,which were not covered in the above mentioned work.