  We report on a theoretical study of $\nu=2$ bilayer quantum Hall systems with
a magnetic field that has a component parallel to the layers. As in the $\nu=1$
case, interlayer phase coherence is closely coupled to electron correlations
and the Aharonov-Bohm phases introduced by a parallel magnetic field can have a
strong influence on the ground state of the system. We find that response of a
$\nu=2$ system to a parallel field is more subtle than that of a $\nu=1$ system
because of the interplay between spin and layer degrees of freedom. There is no
commensurate-incommensurate transition as the parallel field is increased.
Instead, we find a new phase transition which can occur in fixed parallel field
as the interlayer bias potential is varied. The transition is driven by the
competition between canted antiferromagnetic order and interlayer phase
coherence in the presence of the parallel field. We predict a strong
singularity in the differential capacitance of the bilayer which can be used to
detect the phase transition.
