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The 21-cm signal from neutral hydrogen in the early universe will provide unprecedented information about the first stars and galaxies. Extracting this information, however, requires accounting for many unknown astrophysical processes. Semi-numerical simulations are key for exploring the vast parameter space of said processes. These simulations use approximate techniques such as excursion-set and perturbation theory to model the 3D evolution of the intergalactic medium, at a fraction of the computational cost of hydrodynamic and/or radiative transfer simulations. However, exploring the enormous parameter space of the first galaxies can still be computationally expensive. Here we introduce 21cmfish, a Fisher-matrix wrapper for the semi-numerical simulation 21cmFAST. 21cmfish facilitates efficient parameter forecasts, scaling to significantly higher dimensionalities than MCMC approaches, assuming a multi-variate Gaussian posterior. Our method produces comparable parameter uncertainty forecasts to previous MCMC analyses but requires ~10$^4$x fewer simulations. This enables a rapid way to prototype analyses adding new physics and/or additional parameters. We carry out a forecast for HERA using the largest astrophysical parameter space to-date, with 10 free parameters, spanning both population II and III star formation. We find X-ray parameters for the first galaxies could be measured to sub-percent precision, and, though they are highly degenerate, the stellar-to-halo mass relation and ionizing photon escape fraction for population II and III galaxies can be constrained to ~10% precision (logarithmic quantities). Using a principal component analysis we find HERA is most sensitive to the product of the ionizing escape fraction and the stellar-to-halo mass fraction for population II galaxies.