The response of lipid membranes to changes in external pressure is important for many biological processes, and it can also be exploited for technological applications. In this work, we employ all-atom molecular dynamics simulations to characterize the changes in the physical properties of phospholipid bilayers brought about by high pressure (1000 bar). In particular, we study how the response differs, in relation to different chain unsaturation levels, by comparing monounsaturated 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) and biunsaturated dioleoyl-phosphatidylcholine (DOPC) bilayers. Various structural, mechanical, and dynamical features are found to be altered by the pressure increase in both bilayers. Notably, for most properties, including bilayer area and thickness, lipid order parameters, lateral pressure profile, and curvature frustration energy, we observe significantly more pronounced effects for monounsaturated POPC than biunsaturated DOPC. Possible biological implications of the results obtained are discussed, especially in relation to how different lipids can control the structure and function of membrane proteins.