This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter $H ∈ (1/2, 1).$ The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.