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A fast adaptive parallel factor (PARAFAC) decomposition algorithm is proposed for a class of third-order tensors that have one dimension growing linearly with time. It is based on an alternating least squares approach in conjunction with a Newton-type optimization technique. By preserving the Khatri–Rao product and exploiting the reduced-rank update structure of the estimated subspace at each time instant, the algorithm achieves linear complexity and superior convergence performance. A modified version of the algorithm is also proposed to deal with the non-negative constraint. In addition, parallel implementation issues are investigated. Finally, the performance of the algorithm is numerically studied and compared to several state-of-the-art algorithms.