We investigate the dimension-parametric complexity of the reachability problem in vector addition systems with states (VASS) and its extension with pushdown stack (pushdown VASS). Up to now, the problem is known to be F_d-hard for VASS of dimension 3d+2 (the complexity class F_d corresponds to the kth level of the fast-growing hierarchy), and no essentially better bound is known for pushdown VASS. We provide a new construction that improves the lower bound for VASS: F_d-hardness in dimension 2d+3. Furthermore, building on our new insights we show a new lower bound for pushdown VASS: F_d-hardness in dimension d/2 + 6. This dimension-parametric lower bound is strictly stronger than the upper bound for VASS, which suggests that the (still unknown) complexity of the reachability problem in pushdown VASS is higher than in plain VASS (where it is Ackermann-complete).