The complex and computationally expensive nature of landscape evolution models pose significant challenges in the inference and optimisation of unknown parameters. Bayesian inference provides a methodology for estimation and uncertainty quantification of unknown model parameters. In our previous work, we developed parallel tempering Bayeslands as a framework for parameter estimation and uncertainty quantification for the Badlands landscape evolution model. Parallel tempering Bayeslands features high-performance computing with dozens of processing cores running in parallel to enhance computational efficiency. Although we use parallel computing, the procedure remains computationally challenging since thousands of samples need to be drawn and evaluated. \textcolor{black}{In large-scale landscape and basin evolution problems, a single model evaluation can take from several minutes to hours, and in some instances, even days. Surrogate-assisted optimisation has been used for several computationally expensive engineering problems which motivate its use in optimisation and inference of complex geoscientific models.} The use of surrogate models can speed up parallel tempering Bayeslands by developing computationally inexpensive models to mimic expensive ones. In this paper, we apply surrogate-assisted parallel tempering where that surrogate mimics a landscape evolution model by estimating the likelihood function from the model. \textcolor{black}{We employ a neural network-based surrogate model that learns from the history of samples generated. } The entire framework is developed in a parallel computing infrastructure to take advantage of parallelism. The results show that the proposed methodology is effective in lowering the overall computational cost significantly while retaining the quality of solutions.