This paper introduces SPARROW, an efficient forward propagator for Radio Occultation (RO) configurations, capable of simulating electromagnetic wave propagation through both atmospheric and vacuum regions. A shift-map approach is used to reduce the atmospheric computation domain, speeding up simulations compared to traditional Split Step Fourier (SSF) methods. A wavelet-based technique is used for the vacuum propagation step, reducing computational complexity by exploiting wavelet sparsity. Numerical validation shows that SPARROW achieves a high accuracy for extreme atmospheric profiles while optimizing both the computational efficiency and memory usage.