We fulfill a comprehensive study based on density functional theory (DFT) computations to cast insight into the dissociation mechanism of hydrogen molecule on pristine, B-, and N-doped penta-graphene. The doping effect has been also illustrated by varying the concentration of dopant from 4.2 at% (one doping atom in 24 host atoms) to 8.3 at% (two doping atoms in 24 host atoms) and by contemplating different doping sites. Our theoretical investigation shows that the adsorption energy of H-2 molecule and H atom on the substrate can be substantially enhanced by incorporating boron or nitrogen into penta-graphene sheet. The B- and N-doped penta-graphene can effectively decompose H-2 molecule into two H atoms. Our results demonstrate that activation energies for H-2 dissociation and H diffusion on the B- and N-doped penta-graphene are much smaller than the pristine penta-graphene. Further investigation of increasing concentration dopants of the penta-graphene sheet gives sufficiently low activation barrier for H-2 dissociation process. This investigation reveals that the boron and nitrogen dopants can act as effective active site for H-2 dissociation and storage.