Kac and Wakimoto introduced the admissible highest weight representations as a conjectural classification of all modular-invariant representations of the affine Kac–Moody algebras. For the affine Kac–Moody algebra A1(1) their conjectural construction has been proved. Using Kac and Wakimoto's result, Ahn, Chung, and Tye introduced the generalized Fateev–Zamolodchikov parafermionic theories, whose chiral current algebras were recently studied as extensions of the parafermionic vertex operator algebra at admissible level. The characters of the parafermionic vertex operator algebra are string functions of A1(1) up to a simple appropriate factor. For the 1/2-level string functions, we present their mixed mock modular properties as well as elegant mock theta conjecture-like identities involving second-order mock theta functions from Ramanujan's Lost Notebook. In addition, we give an elementary proof that the negative level string functions can be evaluated in terms of false theta functions.