Blind source separation (BSS) using independent component based analysis (e.g., probabilistic ICA and infomax ICA) have been studied in-depth to extract common hemodynamic sources for a group of functional magnetic resonance images (fMRI). The inherent assumption here is that the sources must be non-Gaussian. For most of the real world data, the decomposition is non-unique. Furthermore, there is no quantitative way to determine the component(s) of interest common for the group. This paper shows that using a novel constrained Parallel Factor Analysis (PARAFAC)-based tensor decomposition, one can extract the common task signals and spatial maps from a group of noisy fMRI as rank-1 tensors. The extracted hemodynamic signals have very high correlation with ideal hemodynamic response. A quantitative algorithm to extract common components for a group of subjects is also presented. The modified decomposition preserves the uniqueness under mild conditions which is the most attractive feature for any PARAFAC-based tensor decomposition approach.