Three-term polynomial progressions in subsets of finite fields

Bourgain and Chang recently showed that any subset of ${\mathbb{F}}_p$ of density $\gg p^{-1\mathrm{/}15}$ contains a nontrivial progression $x,x+y,x+y^2$. We answer a question of theirs by proving that if $P_1,P_2\in \mathbb{Z}[y]$ are linearly independent and satisfy $P_1(0)=P_2(0)=0$, then any subset of ${\mathbb{F}}_p$ of density ${\gg }_{P_1,P_2}{p}^{-1\mathrm{/}24}$ contains a nontrivial polynomial progression $x,x+P_1(y),x+P_2(y)$.