Since its introduction as a Maker-Breaker positional game by Duchêne et al. in 2020, the Maker-Breaker domination game has become one of the most studied positional games on vertices. In this game, two players, Dominator and Staller, alternately claim an unclaimed vertex of a given graph G. If at some point the set of vertices claimed by Dominator is a dominating set, she wins; otherwise, i.e. if Staller manages to isolate a vertex by claiming all its closed neighborhood, Staller wins. Given a graph G and a first player, Dominator or Staller must have a winning strategy. We are interested in the computational complexity of determining which player has such a strategy. This problem is known to be PSPACE-complete on bipartite graphs of bounded degree and split graphs; polynomial on cographs, outerplanar graphs, and block graphs; and in NP for interval graphs. In this paper, we consider the parameterized complexity of this game. We start by considering as a parameter the number of moves of both players. We prove that for the general framework of Maker-Breaker positional games in hypergraphs, determining whether Breaker can claim a transversal of the hypergraph in k moves is W[2]-complete, in contrast to the problem of determining whether Maker can claim all the vertices of a hyperedge in k moves, which is known to be W[1]-complete since 2017. These two hardness results are then applied to the Maker-Breaker domination game, proving that it is W[2]-complete to decide if Dominator can dominate the graph in k moves and W[1]-complete to decide if Staller can isolate a vertex in k moves. Next, we provide FPT algorithms for the Maker-Breaker domination game parameterized by the neighborhood diversity, the modular width, the P4-fewness, the distance to cluster, and the feedback edge number.