Simultaneous popular differences for polynomial and linear progressions in finite fields

We study simultaneous popular differences for polynomial progressions and linear progressions over finite fields ${\mathbb{F}}_p$. For certain families of polynomial progressions of algebraic complexity $1$, such as $(x,x+y,x+2y)$ and $(x,x+y^3,x+2y^3)$, we prove the existence of $y\mathrm{\ne }0$ which is popular simultaneously for both progressions. This extends previous results by Conlon, Dong and Hong who considered complexity $0$ progressions. On the other hand, we construct counterexamples showing that simultaneous popular differences are not guaranteed to exist for $(x,x+y,x+2y)$ and $(x,x+y^2,x+2y^2)$, or for $(x,x+y,x+2y)$ and $(x,x+y,x+3y)$, or for $(x,x+y,x+2y)$ and $(x,x+y,x+2y,x+3y)$, answering questions posed by Conlon, Dong and Hong.