The parking-function formula for the second tree inversion enumerator

Let $I_n(q,t)$ and $\widetilde{I}_n(q,t)$ be the two $t$-analogues of the tree inversion enumerator, let $\mathrm{PF}(n)$ denote the set of parking functions of length $n$, and for $\pi\in\mathrm{PF}(n)$ let $\mathrm{oc}(\pi)$ be its parking outcome permutation. Write $\mathrm{cosum}(a)$ for the cosum statistic and $\mathrm{des}(\sigma)$ for the number of descents of a permutation $\sigma$. Parking-function formula. Computational evidence suggests that

$$\widetilde{I}_n(q,t) = \sum_{\pi \in \mathrm{PF}(n)}q^{\mathrm{cosum}(a)}t^{\mathrm{des}(\mathrm{oc}(\pi))}.$$

The analogous formula for $I_n(q,t)$ is known from Stanley and Yin, whereas no such generating-function interpretation was previously available for $\widetilde{I}_n(q,t)$.