The friction between two rubbing surfaces lubricated by water can be diminished if they are coated with phospholipidic bilayers or brushes of polyelectrolytes. In the case of a coating by lipid membranes, the friction is lower when the lipids are in the gel phase rather than in the liquid phase. We investigated the response of fluid or gel bilayers to a mechanical load or under shear using non-equilibrium molecular dynamics simulations (NEMD) to understand whether this difference could come from intermonolayer sliding. The system is composed of a single fully hydrated bilayer of coarse grained phospholipids under a parallel shear with vorticity parallel to the bilayer. In both the liquid and the gel phases, an intermonolayer slip was measured in the velocity profile. In the liquid phase this slip is proportional to the shear stress. In the tilted gel phase of our model the stress is not systematically linear and relaxes differently when the shear is in the direction of the tilt or perpendicular to it. The impact of surface tension (or load) on the friction is different for the liquid and gel phases, but grossly the slip remains of the same order of magnitude.