The low thermal conductivity of phase change materials limits the thermal performance of heat storage devices, hindering their widespread application in engineering. To address this challenge, this paper focused on the fin design of a horizontal triplex-tube latent heat storage device. A novel V-shaped fin with a secondary arrangement is proposed. Numerical simulation is employed to investigate the effects of fin angle, fin length, and heat transfer fluid temperature on the thermal performance of the storage system. Moreover, the fin parameter is optimized by the response surface method. The results demonstrate that, compared to traditional rectangular fins, both the primary and secondary V-shaped fins can significantly enhance the thermal performance. The secondary V-shaped fin has the most significant effects, reducing the melting time of the phase change material by 48.5%, and increasing the average heat storage rate by 83.46%. Additionally, the thermal performance of the heat storage unit exhibits a trend of initially increasing and then decreasing with the increase of fin angle and length, suggesting the existence of optimal structural parameters. Consequently, the optimized secondary V-shaped fin reduces the melting time by 52.9% and increases the average heat storage rate of phase change materials by 104.48%.