Trans-dimensional (trans-D) Bayesian inversion is a powerful tool for estimating seabed geoacoustic models from ocean-acoustic data, combining quantitative model selection with parameter/uncertainty estimation. The approach applies reversible-jump Markov-chain Monte Carlo methods to sample probabilistically over the number of seabed layers and the corresponding geoacoustic parameters for each layer. Layers are added and removed during sampling, referred to as birth and death moves, respectively, changing the dimension of the model. However, the probability of accepting birth and death moves can approach zero for formulations that include many parameters per layer. This paper considers the use of parallel tempering to mitigate this degradation in efficiency. Parallel tempering employs a series of interacting Markov chains with successfully-relaxed acceptance criteria, achieved by raising the likelihood to powers of 1/T, with T greater than or equal to 1 referred to as the sampling temperature. While only the T = 1 chain provides unbiased sampling, probabilistic interchange between chains provides a robust ensemble sampler that mixes more readily over the trans-D model space. The approach is illustrated for wide-angle reflection-coefficient inversion including compressional and shear parameters in the seabed model, resulting in a total of 5 unknown parameters per layer.