Thermal error, which has spatiotemporal behavior, severely reduces machining accuracy of high-accuracy machine tools and should be controlled in real time. The deep learning is used to establish spatiotemporal thermal error model with large-sample thermal information as input. But the acquisition of large-sample thermal information is extremely difficult and costly. So, in the actual application, thermal error is predicted with poor thermal information as input, and then the robustness is weak because the cross-domain and poor spatiotemporal thermal information prediction is still a severe challenge. In this study, the transfer learning model of multi-layer parallel-perceptual-fusion spatiotemporal graph convolutional network is proposed without deepening network depth, instead, its width is expanded. Specifically, the adjacency matrix is constructed to consider the spatial information by defining the distance between each two sensors, and then the graph convolutional network is integrated into long short-term memory and gated recurrent unit to propose graph convolutional long short-term memory and graph convolutional gated recurrent unit with constructed adjacency matrix as input, respectively. The graph convolutional long short-term memory and graph convolutional gated recurrent unit are superimposed to propose the multi-layer parallel-perceptual-fusion spatiotemporal graph convolutional network. To improve the robustness, the adjacency matrix is retrained, and the transfer learning model of multi-layer parallel-perceptual-fusion spatiotemporal graph convolutional network is proposed and applied to forecast thermal error, and its prediction accuracy is 94.926%. Spatiotemporal characteristics of temperature and thermal error are fully captured by transfer learning model of graph convolutional long short-term memory and graph convolutional gated recurrent unit. Finally, the transfer learning model of multi-layer parallel-perceptual-fusion spatiotemporal graph convolutional network is embedded into thermal error control service architecture based on cloud-edge collaboration. With implementation of thermal error control service architecture based on cloud-edge collaboration, machining error of machine tools is reduced by more than 80%.