Planar Turán number $ex_{\mathcal{P}}(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the planar Turán number for double stars. In this paper, we prove that $ex_{\mathcal{P}}(n,S_{2,4})\leq \frac{31}{14}n$ for $n\geq 1$, and show that equality holds for infinitely many integers $n$.