In this paper it is shown that the one-dimensional configuration sums of the
solvable lattice models of Andrews, Baxter and Forrester and the string functions
associated with admissible representations of the affine Lie algebra A$_1^{(1)}$ as
introduced by Kac and Wakimoto can be exploited to yield a very general class of conjugate
Bailey pairs. Using the recently established fermionic or constant-sign expressions for the
one-dimensional configuration sums, our result is employed to derive fermionic expressions
for fractional-level string functions, parafermion characters and A$_1^{(1)}$ branching
functions. In addition, $q$-series identities are obtained whose Lie algebraic and/or
combinatorial interpretation is still lacking.