Bone marrow and adipose tissue human mesenchymal stem cells were seeded in highly performing 3D gelatin–chitosan hybrid hydrogels of varying chitosan content in the presence of human platelet lysate and evaluated for their proliferation and osteogenic differentiation. Both bone marrow and adipose tissue human mesenchymal stem cells in gelatin–chitosan hybrid hydrogel 1 (chitosan content 8.1%) or gelatin–chitosan hybrid hydrogel 2 (chitosan 14.9%) showed high levels of viability (80%–90%), and their proliferation and osteogenic differentiation was significantly higher with human platelet lysate compared to fetal bovine serum, particularly in gelatin–chitosan hybrid hydrogel 1. Mineralization was detected early, after 21 days of culture, when human platelet lysate was used in the presence of osteogenic stimuli. Proteomic characterization of human platelet lysate highlighted 59 proteins mainly involved in functions related to cell adhesion, cellular repairing mechanisms, and regulation of cell differentiation. In conclusion, the combination of our gelatin–chitosan hybrid hydrogels with hPL represents a promising strategy for bone regenerative medicine using human mesenchymal stem cells.