  Given an algebraic family $f\colon\Lambda\times \mathbb{A}^2 \to
\Lambda\times \mathbb{A}^2$ of plane polynomial automorphisms of H\'enon 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 $Y\subset \Lambda \times \mathbb{A}^2 \to \Lambda$. 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 $Y_t, t\in \Lambda$, when $Y$ is non-degenerate. Then we study in more
details the case of dissipative families of quadratic H\'enon maps.
