  We study integrable and conformal boundary conditions for ^sl(2) Z_k
parafermions on a cylinder. These conformal field theories are realized as the
continuum scaling limit of critical A-D-E lattice models with negative spectral
parameter. The conformal boundary conditions labelled by (a,m) in (G, Z_{2k})
are identified with associated integrable lattice boundary conditions labelled
by (r,a) in (A_{g-2},G) where g is the Coxeter number of the A-D-E graph G. We
obtain analytically the boundary free energies, present general expressions for
the parafermion cylinder partition functions and confirm these results by
numerical calculations.
