Given an algebraic family $f\colonΛ\times \mathbb{A}^2 \to Λ\times \mathbb{A}^2$ of plane polynomial automorphisms of Hénon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K}$, we study the number of periodic points contained in a family of subvarieties $Y\subset Λ\times \mathbb{A}^2 \to Λ$. Under some mild conditions (e.g., when the family is dissipative), we show that there are only finitely many periodic points contained in a non periodic marked point. This generalizes a result of Charles Favre and Romain Dujardin. We also show that there is a uniform bound on the number of periodic points in a family of curves $Y_t, t\in Λ$, when $Y$ is non-degenerate. Then we study in more details the case of dissipative families of quadratic Hénon maps.