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We present an analytical description of localized spin wave modes that form in a parabolic field minimum in a thin ferromagnetic film. Mode profiles proportional to Hermite functions are eigenfuctions of the applied field and exchange parts of the equations of motion, and also provide a basis for numerical approximation of magnetostatic interactions. We find that the spin wave modes are roughly equally spaced in frequency and have roughly equal coupling to a uniform driving field. The calculated mode frequencies and corresponding profiles of localized spin wave modes are in good agreement with micromagnetic modeling and previously published experimental results on multiple resonances from a series of localized modes detected by ferromagnetic resonance force microscopy.