High-resolution spectral estimation is an important subject in many applications of modern signal processing. The fundamental problem in applying various high-resolution spectral estimation algorithms is the computational complexity. Recently, the truncated QR methods have been shown to be comparable to the SVD- based methods for the sinusoidal frequency estimation based on the forward-backward linear prediction (FBLP) model. However, without exploiting the special structure of the FBLP matrix, the QR decomposition (QRD) of the FBLP matrix has the computational complexity on the order of n cubic for a 2m x n FBLP matrix. Here we propose a fast algorithm to perform the QRD of the FBLP matrix. It is based on exploiting the special Toeplitz-Hankel form of the FBLP matrix. The computational complexity is then reduced to the order of n square. The fast algorithm can also be easily implemented onto a linear systolic array. The number of time steps required is further reduced to 2m + 5n - 4 by using the parallel implementation. The geometric transformation, which improves the numerical stability, for the downdating of the Cholesky factors is also considered.