Multistage stochastic optimization leads to NLPs over scenario trees that become extremely large when many time stages or fine discretizations of the probability space are required. Interior-point methods are well suited for these problems if the arising huge, structured KKT systems can be solved efficiently, for instance, with a large scenario tree but a moderate number of variables per node. For this setting we develop a distributed implementation based on data parallelism in a depth-first distribution of the scenario tree over the processes. Our theoretical analysis predicts very low memory and communication overheads. Detailed computational experiments confirm this prediction and demonstrate the overall performance of the algorithm. We solve multistage stochastic quadratic programs with up to 400e6 variables and 8.59e9 KKT matrix entries or 136e6 variables and 12.6e9 entries on a compute cluster with 384 GiB of RAM.