$p$-Dominating-Set is W[2]-complete under fpt-reductions [FG06, Corollary 7.15]: it is fpt-equivalent to $p$-Hitting-Set [FG06, Example 2.7], which is W[2]-complete ($WH_E2_HittingSetW2Complete$).

• Dominating Set $\le$ Hitting Set. The universe is the vertex set, and the hyperedges are the closed neighbourhoods $N[v]$: a set dominates the graph exactly when it meets every $N[v]$.
• Hitting Set $\le$ Dominating Set. The graph has a vertex for every element and every hyperedge; the elements form a clique, and an element is joined to the hyperedges containing it. A hitting set of $k$ elements dominates the graph; conversely, a dominating set of $k$ vertices yields a hitting set of $k$ elements by replacing each hyperedge-vertex by an element of that hyperedge. The universe is first restricted to the elements occurring in the sets; instances with an empty hyperedge or with $k$ larger than the universe are mapped to a fixed no-instance.