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Tensor regression models, such as CP regression and Tucker regression, have many successful applications in neuroimaging analysis where the covariates are of ultrahigh dimensionality and possess complex spatial structures. The high-dimensional covariate arrays, also known as tensors, can be approximated by low-rank structures and fit into the generalized linear models. The resulting tensor regression achieves a significant reduction in dimensionality while remaining efficient in estimation and prediction. Brain functional connectivity is an essential measure of brain activity and has shown significant association with neurological disorders such as Alzheimer's disease. The symmetry nature of functional connectivity is a property that has not been explored in previous tensor regression models. In this work, we propose a sparse symmetric tensor regression that further reduces the number of free parameters and achieves superior performance over symmetrized and ordinary CP regression, under a variety of simulation settings. We apply the proposed method to a study of Alzheimer's disease (AD) and normal ageing from the Berkeley Aging Cohort Study (BACS) and detect two regions of interest that have been identified important to AD.