Continual table semantic parsing aims to train a parser on a sequence of tasks, where each task requires the parser to translate natural language into SQL based on task-specific tables but only offers limited training examples. Conventional methods tend to suffer from overfitting with limited supervision, as well as catastrophic forgetting due to parameter updates.Despite recent advancements that partially alleviate these issues through semi-supervised data augmentation and retention of a few past examples, the performance is still limited by the volume of unsupervised data and stored examples.To overcome these challenges, this paper introduces a novel method integrating parameter-efficient fine-tuning (PEFT) and in-context tuning (ICT) for training a continual table semantic parser. Initially, we present a task-adaptive PEFT framework capable of fully circumventing catastrophic forgetting, which is achieved by freezing the pre-trained model backbone and fine-tuning small-scale prompts. Building on this, we propose a teacher-student framework-based solution. The teacher addresses the few-shot problem using ICT, which procures contextual information by demonstrating a few training examples. In turn, the student leverages the proposed PEFT framework to learn from the teacher's output distribution, and subsequently compresses and saves the contextual information to the prompts, eliminating the need to store any training examples.Experimental evaluations on two benchmarks affirm the superiority of our method over prevalent few-shot and continual learning baselines across various metrics.