  Let $G$ be a graph on $n$ vertices and $\mathrm{STAB}_k(G)$ be the convex
hull of characteristic vectors of its independent sets of size at most $k$. We
study extension complexity of $\mathrm{STAB}_k(G)$ with respect to a fixed
parameter $k$ (analogously to, e.g., parameterized computational complexity of
problems). We show that for graphs $G$ from a class of bounded expansion it
holds that $\mathrm{xc}(\mathrm{STAB}_k(G))\leqslant \mathcal{O}(f(k)\cdot n)$
where the function $f$ depends only on the class. This result can be extended
in a simple way to a wide range of similarly defined graph polytopes. In case
of general graphs we show that there is {\em no function $f$} such that, for
all values of the parameter $k$ and for all graphs on $n$ vertices, the
extension complexity of $\mathrm{STAB}_k(G)$ is at most $f(k)\cdot
n^{\mathcal{O}(1)}.$ While such results are not surprising since it is known
that optimizing over $\mathrm{STAB}_k(G)$ is $FPT$ for graphs of bounded
expansion and $W[1]$-hard in general, they are also not trivial and in both
cases stronger than the corresponding computational complexity results.
