This study presents a stochastic parallel gradient descent (SPGD) algorithm based on the optimization of square of wavefront aberration gradient to improve the convergence speed of the SPGD algorithm for correcting the wavefront aberrations. Based on the principle of the SPGD algorithm, this study analyzes the constrained factors of the convergence speed in the algorithm iteration. Further, the linear relation between the square of wavefront aberration gradient and disturbances from random aberration is analyzed by deriving the theoretical formula; subsequently, the square of wavefront aberration gradient is approximately calculated by using the far-field spot's normalized second moment, and the optical correction of wavefront aberrations is finally realized. The convergence speed and correction effect of the proposed SPGD algorithm are then analyzed using numeric simulations and compared with those of the previous SPGD algorithms. Finally, a wavefront sensorless adaptive optics correction experiment for Fresnel zone plate wavefront aberrations is performed to validate the performance of the proposed SPGD algorithm. The numeric simulation and results of correction experiments consistently denote that the proposed method has a high convergence speed and a robust corresponding adaptability.