  We present a parameterized dichotomy for the \textsc{$k$-Sparsest Cut}
problem in weighted and unweighted versions. In particular, we show that the
weighted \textsc{$k$-Sparsest Cut} problem is NP-hard for every $k\geq 3$ even
on graphs with bounded vertex cover number. Also, the unweighted
\textsc{$k$-Sparsest Cut} problem is W[1]-hard when parameterized by the three
combined parameters tree-depth, feedback vertex set number, and $k$. On the
positive side, we show that unweighted \textsc{$k$-Sparsest Cut} problem is FPT
when parameterized by the vertex cover number and $k$, and when $k$ is fixed,
it is FPT with respect to the treewidth. Moreover, we show that the generalized
version \textsc{$k$-Small-Set Expansion} problem is FPT when parameterized by
$k$ and the maximum degree of the graph, though it is W[1]-hard for each of
these parameters separately.
