Abstract Meaning Representations (AMR) are a broad-coverage semantic formalism which represents sentence meaning as a directed acyclic graph. To train most AMR parsers, one needs to segment the graph into subgraphs and align each such subgraph to a word in a sentence; this is normally done at preprocessing, relying on hand-crafted rules. In contrast, we treat both alignment and segmentation as latent variables in our model and induce them as part of end-to-end training. As marginalizing over the structured latent variables is infeasible, we use the variational autoencoding framework. To ensure end-to-end differentiable optimization, we introduce a differentiable relaxation of the segmentation and alignment problems. We observe that inducing segmentation yields substantial gains over using a `greedy' segmentation heuristic. The performance of our method also approaches that of a model that relies on the segmentation rules of \citet{lyu-titov-2018-amr}, which were hand-crafted to handle individual AMR constructions.