We study semi-supervised sequence prediction tasks where labeled data are too scarce to effectively finetune a model and at the same time few-shot prompting of a large language model (LLM) has suboptimal performance. This happens when a task, such as parsing, is expensive to annotate and also unfamiliar to a pretrained LLM. In this paper, we present a discovery that student models distilled from a prompted LLM can often generalize better than their teacher on such tasks. Leveraging this finding, we propose a new distillation method, multistage collaborative knowledge distillation from an LLM (MCKD), for such tasks. MCKD first prompts an LLM using few-shot in-context learning to produce pseudolabels for unlabeled data. Then, at each stage of distillation, a pair of students are trained on disjoint partitions of the pseudolabeled data. Each student subsequently produces new and improved pseudolabels for the unseen partition to supervise the next round of student(s) with. We show the benefit of multistage cross-partition labeling on two constituency parsing tasks. On CRAFT biomedical parsing, 3-stage MCKD with 50 labeled examples matches the performance of supervised finetuning with 500 examples and outperforms the prompted LLM and vanilla KD by 7.5% and 3.7% parsing F1, respectively.