Enhancement of polymer thermal conductivity using nanographene fillers and clarification of its molecular-scale mechanisms are of great concern in the development of advanced thermal management materials. In the present study, molecular dynamics simulation was employed to theoretically show that the in-plane aspect ratio of a graphene filler can have a significant impact on the effective thermal conductivity of paraffin/graphene composites. Our simulation included multiple graphene fillers aggregated in a paraffin matrix. The effective thermal conductivity of a paraffin/graphene composite, described as a second-rank tensor in the framework of equilibrium molecular dynamics simulation, was calculated for two types of graphene fillers with the same surface area but in-plane aspect ratios of 1 and 10. The filler with the higher aspect ratio was found to exhibit a much higher thermal conductivity enhancement than the one with the lower aspect ratio. This is because a high in-plane aspect ratio strongly restricts the orientation of fillers when they aggregate and, consequently, highly ordered agglomerates are formed. On decomposing the effective thermal conductivity tensor into various molecular-scale contributions, it was identified that the thermal conductivity enhancement is due to the increased amount of heat transfer inside the graphene filler, particularly along the longer in-plane axis. The present result indicates a possibility of designing the heat conduction characteristics of a nanocomposite by customizing the filler shapes so as to control the aggregation structure of the fillers.