An analytical and adaptive parameter extraction method is proposed to accurately and efficiently extract the parameters of solar cells' triple-diode lumped-parameter equivalent circuit model. This method can obtain explicit expressions of parameters, facilitating seamless integration into photovoltaic systems for circuit-level simulation and design. The proposed method addresses the lack of analytical parameter extraction methods for triple-diode model, avoids the iteration of numerical methods, and significantly reduces computational complexity, thus improving the overall efficiency of the modeling process. In this method, the principle of effective-diode-based analysis is used to simplify the complex multi-diode model. Recursive thinking is then combined to adaptively extract all parameters through multiple steps according to the number of diodes in the model. Subsequently, the practicability of this method is evaluated by testing on three different equivalent circuit models of solar cells including the single-, double-, and triple-diode models. Furthermore, the universality of this method is demonstrated by testing on three photovoltaic modules of different types including Mono-crystalline, Multi-crystalline, Thin-film, and perovskite solar cells at different temperatures and irradiances. Finally, the simulation results based on the extracted parameters are compared with the experimental data to calculate the root mean square error (RMSE) magnitude, which ranges from 10-5 to 10-3, demonstrating the feasibility of the proposed method. As a result, this analytical parameter extraction method provides a robust tool and approach for research and applications in the field of solar cell modeling.