  Consider a uniform rooted Cayley tree $T_{n}$ with $n$ vertices and let $m$
cars arrive sequentially, independently, and uniformly on its vertices. Each
car tries to park on its arrival node, and if the spot is already occupied, it
drives towards the root of the tree and parks as soon as possible. Lackner &
Panholzer (arXiv:1504.04972) established a phase transition for this process
when $ m \approx \frac{n}{2}$. In this work, we couple this model with a
variant of the classical Erd\"os-R\'enyi random graph process. This enables us
to describe the phase transition for the size of the components of parked cars
using a modification of the multiplicative coalescent which we name the frozen
multiplicative coalescent. The geometry of critical parked clusters is also
studied. Those trees are very different from Bienaym\'e-Galton-Watson trees and
should converge towards the growth-fragmentation trees canonically associated
to the $3/2$-stable process that already appeared in the study of random planar
maps.
