  We construct two-dimensional conformal field theories with a Z_N symmetry,
based on the second solution of Fateev-Zamolodchikov for the parafermionic
chiral algebra. Primary operators are classified according to their
transformation properties under the dihedral group (Z_N x Z_2, where Z_2 stands
for the Z_N charge conjugation), as singlets, [(N-1)/2] different doublets, and
a disorder operator. In an assumed Coulomb gas scenario, the corresponding
vertex operators are accommodated by the Kac table based on the weight lattice
of the Lie algebra B_{(N-1)/2} when N is odd, and D_{N/2} when N is even. The
unitary theories are representations of the coset SO_n(N) x SO_2(N) /
SO_{n+2}(N), with n=1,2,.... We suggest that physically they realize the series
of multicritical points in statistical systems having a Z_N symmetry.
