The interaction between nucleic acid bases and graphene, along with its derivatives, is crucial for various applications like biosensors, drug delivery, and bio-nanotechnology. The present study focuses on optimizing nucleic acid bases, and the graphene layer using the B3LYP/6-31G** method and the beryllium-oxide-graphene layer using the PBE/6-31G** method. The optimized coordinates are then used to investigate the interaction of nucleic acid bases with pristine graphene and beryllium-oxide-graphene in parallel stacking mode. Various functionals including hybrid B3LYP, PBEPBE, and range-separated ωB97XD along with 6-31G(d,p) basis sets, are employed for this analysis. Based on the obtained interaction energy values, it is concluded that the wB97XD/6-31G(d,p) method provides superior results for the interaction study. Additionally, it is noted that the interaction with pristine graphene primarily involves aromatic π-π interactions and hydrogen bonds, while beryllium oxide graphene seems to form coordination bonds with electron-rich sites. Also, the beryllium-oxide-graphene layer exhibited reduced rigidity compared to pristine carbon graphene layer.