The relationship between Catalan combinatorics, the representation theory of the elliptic Hall algebra, and knot homology has been extensively studied over the past two decades. Building on this work, we prove that the symmetric function $e_{(1^k)}[-MX^{m,n}] \cdot 1$, arising from the elliptic Hall algebra, equals the generating function for $k$-tuples of cyclic $(m,n)$-parking functions. This result resolves a conjecture of Gorsky--Mazin--Vazirani and Wilson, establishing that the elliptic Hall algebra governs the Khovanov--Rozansky homology of torus links $T(km,kn)$. Consequently, this provides an affirmative answer to a question posed by Galashin and Lam in the torus link case. As a key step in the proof, we develop a rational analogue of the Shareshian--Wachs involution, which was originally introduced to prove the symmetry property of chromatic quasisymmetric functions. This is based on a cowork with Donghyun Kim.