  We study parity games in which one of the two players controls only a small
number $k$ of nodes and the other player controls the $n-k$ other nodes of the
game. Our main result is a fixed-parameter algorithm that solves bipartite
parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in
time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct
priorities and $m$ is the number of edges. For all games with $k = o(n)$ this
improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and
Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved
deterministic algorithm for graphs with small average degree.
