Tensors, or multidimensional data arrays, require dimension reduction in modeling applications due to their large size. In addition, these structures typically exhibit inherent sparsity, requiring the use of regularization methods to properly characterize an association between a covariate and a response. We propose a Bayesian method to parsimoniously model a scalar response with a tensor-valued covariate using the Tucker tensor decomposition. This method retains the spatial relationship within an image covariate, while reducing the number of parameters varying within the model and applying appropriate regularization methods. Simulated data are analyzed to demonstrate model effectiveness, with comparisons made to both classical and Bayesian methods. A neuroimaging analysis using data from the Alzheimer's Data Neuroimaging Initiative is performed to demonstrate the method.