Recent years have witnessed the success of applying seq2seq models to graph parsing tasks, where the outputs are compositionally structured (e.g., a graph or a tree). However, these seq2seq approaches pose a challenge in quantifying the model’s compositional uncertainty on graph structures due to the gap between seq2seq output probability and structural probability on the graph. This work is the first to quantify and evaluate compositional uncertainty for seq2seq graph parsing tasks. First, we proposed a generic, probabilistically interpretable framework that allows correspondences between seq2seq output probability to structural probability on the graph. This framework serves as a powerful medium for quantifying a seq2seq model's compositional uncertainty on graph elements (i.e., nodes or edges). Second, to evaluate uncertainty quality in terms of calibration, we propose a novel metric called Compositional Expected Calibration Error (CECE) which can measure a model’s calibration behavior in predicting graph structures. By a thorough evaluation for compositional uncertainty on three different tasks across ten domains, we demonstrate that CECE is a better reflection for distributional shift compared to vanilla sequence ECE. Finally, we validate the effectiveness of compositional uncertainty considering the task of collaborative semantic parsing, where the model is allowed to send limited subgraphs for human review. The results show that the collaborative performance based on uncertain subgraph selection consistently outperforms random subgraph selection (30% average error reduction rate) and performs comparably to oracle subgraph selection (only 0.33 difference in average prediction error), indicating that compositional uncertainty is an ideal signal for model errors and can benefit various downstream tasks.