  Given an undirected graph $G$ and a multiset of $k$ terminal pairs
$\mathcal{X}$, the Vertex-Disjoint Paths (\VDP) and Edge-Disjoint Paths (\EDP)
problems ask whether $G$ has $k$ pairwise internally vertex-disjoint paths and
$k$ pairwise edge-disjoint paths, respectively, connecting every terminal pair
in~$\mathcal{X}$. In this paper, we study the kernelization complexity of
\VDP~and~\EDP~on subclasses of chordal graphs. For \VDP, we design a $4k$
vertex kernel on split graphs and an $\mathcal{O}(k^2)$ vertex kernel on
well-partitioned chordal graphs. We also show that the problem becomes
polynomial-time solvable on threshold graphs. For \textsc{EDP}, we first prove
that the problem is $\mathsf{NP}$-complete on complete graphs. Then, we design
an $\mathcal{O}(k^{2.75})$ vertex kernel for \EDP~on split graphs, and improve
it to a $7k+1$ vertex kernel on threshold graphs. Lastly, we provide an
$\mathcal{O}(k^2)$ vertex kernel for \EDP~on block graphs and a $2k+1$ vertex
kernel for clique paths. Our contributions improve upon several results in the
literature, as well as resolve an open question by Heggernes et al.~[Theory
Comput. Syst., 2015].
