Continual Semantic Parsing (CSP) aims to train parsers to convert natural language questions into SQL across tasks with limited annotated examples, adapting to the real-world scenario of dynamically updated databases. Previous studies mitigate this challenge by replaying historical data or employing parameter-efficient tuning (PET), but they often violate data privacy or rely on ideal continual learning settings. To address these problems, we propose a new Large Language Model (LLM)-Enhanced Continuous Semantic Parsing method, named LECSP, which alleviates forgetting while encouraging generalization, without requiring real data replay or ideal settings. Specifically, it first analyzes the commonalities and differences between tasks from the SQL syntax perspective to guide LLMs in reconstructing key memories and improving memory accuracy through a calibration strategy. Then, it uses a task-aware dual-teacher distillation framework to promote the accumulation and transfer of knowledge during sequential training. Experimental results on two CSP benchmarks show that our method significantly outperforms existing methods, even those utilizing data replay or ideal settings. Additionally, we achieve generalization performance beyond the upper limits, better adapting to unseen tasks.