We propose an alternating minimization (AM) algorithm for estimating attenuation functions in X-ray transmission tomography using priors that promote sparsity in the pixel/voxel differences domain. As opposed to standard maximum-a-posteriori (MAP) estimation, we use the automatic relevance determination (ARD) framework. In the ARD approach, sparsity (or compressibility) is promoted by introducing latent variables which serve as the weights of quadratic penalties, with one weight for each pixel/voxel; these weights are then automatically learned from the data. This leads to an algorithm where the quadratic penalty is reweighted in order to effectively promote sparsity. In addition to the usual object estimate, ARD also provides measures of uncertainty (posterior variances) which are used at each iteration to automatically determine the trade-off between data fidelity and the prior, thus potentially circumventing the need for any tuning parameters. We apply the convex decomposition lemma in a novel way and derive a separable surrogate function that leads to a parallel algorithm. We propose an extension of branchless distance-driven forward/back-projections which allows us to considerably speed up the computations associated with the posterior variances. We also study the acceleration of the algorithm using ordered subsets.