Reading the abstract for the paper (thanks to skybrian for providing the link) it seems that the phrase "anti-aging vaccine" is a gross exaggeration in this context rather than the science itself being flawed.

Broadly, the abstract seems to be describing a particular cell-surface marker found to be associated with senescence ("seno-antigen") which can be targeted by a therapy ("genetic ablation") which accelerates removal of cells with that marker. Doing so improves the lifespan of mice so treated.

Thus, as the abstract concludes, "...vaccination targeting seno-antigens could be a potential strategy for new senolytic therapies."