  Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical
type. We show that periodic elements of $A$ and the elements preserving some
parabolic subgroup of $A$ act elliptically on the additional length graph
$\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$
constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically
hyperbolic. We use these results to find an element $g\in A$ such that $\langle
P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic
subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the
Garside generators, independently of $A$. This allows us to show that, in
contrast with the Artin generators case, the sequence
$\{\omega(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of
braid groups with respect to the Garside generating set, goes to infinity.
