Foundation models have superior performance across a wide array of machine learning tasks. The training of these models typically involves model parallelism (MP) to navigate the constraints of GPU memory capacity. However, MP strategies involve transmitting model activations between GPUs, which can hinder training speed in large clusters. Previous research has examined gradient compression in data-parallel contexts, but its applicability in MP settings remains largely unexplored. In this paper, we investigate the unique characteristics of compression in MP and study why strategies from gradient compression might not be directly applicable to MP scenarios. Subsequently, to systematically understand the capabilities and limitations of \underline{M}odel Parallelism \underline{C}ompression, we present a benchmarking framework \textbf{MCBench}. MCBench not only includes four major categories of compression algorithms but also includes several widely used models spanning language and vision tasks on a well-established distributed training framework, Megatron-LM. We initiate the first comprehensive empirical study by using MCBench. Our empirical study encompasses both the fine-tuning and pre-training of FMs. We probe over 200 unique training configurations and present results using 10 widely used datasets. To comprehend the scalability of compression advantages with the expansion of model size and cluster size, we propose a novel cost model designed specifically for training with MP compression. The insights derived from our findings can help direct the future development of new MP compression algorithms for distributed training.