Optical pulling along straight trajectories has been successfully demonstrated for both dipolar and Mie particles using optical gradient and/or scattering forces over the past decade. However, much less attention is devoted to the pulling along curved paths, particularly for Mie particles, since the mechanism of continuous attraction based on intensity gradients is effective only for dipolar particles, thus limiting its practical applications. Here, we demonstrate the optical pulling of Mie particles with gain along a parabolic trajectory immersed in a two-dimensional vector Airy beam. Numerical results based on the decomposed force expressions exhibit that the longitudinal optical pulling effect is overwhelmingly dominated by the optical scattering force, while the optical gradient force has a negligible contribution. In addition, the Mie particle is shown to be transversely trapped stably in the main lobe of the Airy beam via the transverse gradient force in favor of the realization of optical pulling. Such an optical pulling phenomenon can also be observed for a gold particle coated with a suitable gain shell. Our findings may provide new insights into the customization of optical pulling in curved paths.