Parent-identifying set system is a kind of combinatorial structures with applications to broadcast encryption. In this paper we investigate the maximum number of blocks $I_2(n,4)$ in a $2$-parent-identifying set system with ground set size $n$ and block size $4$. The previous best known lower bound states that $I_2(n,4)=Ω(n^{4/3+o(1)})$. We improve this lower bound by showing that $I_2(n,4)= Ω(n^{3/2-o(1)})$ using techniques in additive number theory.