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Optical fibers with index profiles that are optimized for broadband double pumped parametric amplifiers are thoroughly investigated. The analysis includes random variations in the radii of the profile along the fiber length (introduced in the fabrication process). By optimizing the fourth order dispersion parameter and the effective area we obtain fiber designs that provide flat spectral gain over 200 nm with 3.5 W pump lasers and, at the same time, have robust operation wavelength against fluctuations of up to 1% in the radii of the index profile. These fibers show positive and very small fourth order dispersion. Our results indicate that in such low dispersion fibers, the sixth order dispersion parameter must be considered in the optimization algorithm.