  We investigate the question of parity breaking in three-dimensional Euclidean
SU(2) gauge-Higgs theory by Monte Carlo simulations. We observe no sign of
spontaneous parity breaking in the behaviour of both local and non-local gauge
invariant operators. However, the presence of parity odd terms in the action
can induce a phase transition to a parity odd ground state which is
characterized by a Chern-Simons like condensate. The implications for various
proposed scenarios of fermion number non-conservation is discussed.
