In this paper we construct new parafermion theories, generalizing the ZL parafermion theory from integer L to rational L. These non-unitary parafermion theories (which are also defined as SL(2, R)L/U(1)) have some novel features: an infinite number of currents with negative conformal dimensions for most (if not all) of them. We construct the string functions of these new parafermion theories. Generalizing Felder's BRST cohomology approach we construct from the string functions the branching functions of the SU(2)L × SU(2)K SU(2)K+L coset theories, where both K, L are rational. New N = 2 superconformal field theories and topological field theories are also constructed. Their characters are obtained in terms of the new string functions.