This paper deals with anisotropic mesh adaptation applied to unsteady inviscid CFD simulations. Anisotropic metric-based mesh adaptation is an efficient strategy to reduce the extensive CPU time currently required by time-dependent simulations from the perspective of performing this kind of computations on a daily basis in an industrial context. In this work, we detail the time-accurate extension of multi-scale anisotropic mesh adaptation for steady flows, i.e., a control of the interpolation error in Lp norm, to unsteady flows based on a space-time error analysis and an enhanced version of the fixed-point algorithm. We also show that each stage - remeshing, metric field computation, solution transfer, and flow solution - is important to design an efficient time-accurate anisotropic mesh adaptation process. The parallelization of the whole mesh adaptation platform is also discussed. The efficiency of the approach is emphasized on three-dimensional problems with convergence analysis and CPU data.