  We construct a parafermionic conformal theory with the symmetry Z_N, for N
odd, based on the second solution of Fateev-Zamolodchikov for the corresponding
parafermionic chiral algebra. Primary operators are classified according to
their transformation properties under the dihedral group D_N, as singlet,
doublet 1,2,...,(N-1)/2, and disorder operators. In an assumed Coulomb gas
scenario, the corresponding vertex operators are accommodated by the weight
lattice of the Lie algebra B_(N-1)/2. The unitary theories are representations
of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,... . Physically, they
realise the series of multicritical points in statistical theories having a D_N
symmetry.
