Fast Fourier transform (FFT) algorithms based on the Cooley-Tukey approach [cf. \textit{J. W. Cooley} and \textit{J. W. Tukey}, Math. Comput. 19, 297-301 (1965; Zbl 0127.090)] are developed for single instruction multiple data (SIMD) machines. Any combination of FFT with powers of two as the periods and some alignment requirements for the initial data may be calculated. The algorithm solves multiple instances of similar problems with a degree of parallelism, therefore it is called superparallel in the sense of \textit{D. Parkinson} [Superparallel algorithms, in Supercomputing, NATO ASI series \(F\), Vol. 62, Springer-Verlag, Berlin, New York (1989)]. The degree of parallelism is proportional to \(\log_ 2m\), where \(m\) is the number of points in the largest subsystem. Unscrambling the results may be scheduled in parallel if necessary. An implementation for the SIMD machine MasPar with 16,384 processors is analysed.