Deep learning (DL) jobs use multi-dimensional parallelism, i.e. combining data, model, and pipeline parallelism, to use large GPU clusters efficiently. Long-running jobs may experience changes to their GPU allocation: (i) resource elasticity during training adds or removes GPUs; (ii) hardware maintenance may require redeployment on different GPUs; and (iii) GPU failures force jobs to run with fewer devices. Current DL frameworks tie jobs to a set of GPUs and thus lack support for these scenarios. In particular, they cannot change the multi-dimensional parallelism of an already-running job in an efficient and model-independent way. We describe Tenplex, a state management library for DL systems that enables jobs to change their parallelism dynamically after the GPU allocation is updated at runtime. Tenplex achieves this through a new abstraction, a parallelizable tensor collection (PTC), that externalizes the job state during training. After a GPU change, Tenplex uses the PTC to transform the job state: the PTC repartitions the dataset state under data parallelism and exposes it to DL workers through a virtual file system; and the PTC obtains the model state as partitioned checkpoints and transforms them to reflect the new parallelization configuration. For efficiency, Tenplex executes PTC transformations in parallel with minimum data movement between workers. Our experiments show that Tenplex enables DL jobs to support dynamic parallelization with low overhead.