We investigate diffusion of a peptide drug through Graphene Oxide (GO) membranes that are modeled as a porous layered laminate constructed from aligned flakes of GO. Our experiments using a peptide drug show a tunable non-linear dependence of the peptide concentration upon time. This is confirmed using numerical simulations with a diffusion equation accounting for the photothermal degradation of fluorophores and an effective percolation model. This modeling yields an interpretation of the control and delay of drug diffusion through GO membranes. The ability to modulate the density of hydrogel-like GO membranes to control drug release rates could be a step forwards in tailoring drug release properties of the hydrogels for therapeutic applications.