It you use an explict scheme as you describe for the hyperbolic part (but implicit for the viscous part) with small timestep satisfying a CFL condition, you could hopefully parallelize both the linear system to solve (viscous part) and the explicit part. But you would nevertheless get oscillations. To avoid oscillations you need either to use specific schemes for the hyperbolic part, involving approximate Riemann solvers, or use a fine enough mesh (as described in my previous message) so that at the mesh level the viscosity smoothes out the oscillations. If Re is moderate and Ma is not too small this last option is excellent because in this case hopefully you do not need to satisfy any CFL condition.