Generation and propagation of optical skyrmions provide a versatile plalform for topologically nontrivial optical informatics and light-matter interactions, but their acceleration along curved trajectories is to be studied. In this study, we experimentally demonstrate the first accelerating skyrmion lattices conveyed by Airy structured light, characterized by topologically stable skyrmion textures with self-acceleration along parabolic trajectories. We show that the skyrmion unit cell can maintain a Skyrme number $|N_\text{sk}|>0.9$ within a propagation range of $\pm1.22\ z_R$ upon parabolic acceleration. Notably, the meron structure remains $|N_\text{sk}|$ stable within $0.5\pm0.02$ over a significantly extended range of $\pm3.06\ z_R$. Our work provides a new potential carrier for topologically robust information distribution, particle sorting and manipulation.