  Consider a supercritical Bienaym\'e--Galton--Watson tree $ \mathcal{T}$ with
geometric offspring distribution. Each vertex of this tree represents a parking
spot which can accommodate at most one car. On the top of this tree, we add
$(A_u : u \in \mathcal{T})$ i.i.d.\ non negative integers sampled according to
a given law $ \mu$, which are the car arrivals on $ \mathcal{T}$. Each car
tries to park on its arriving vertex and if the spot is already occupied, it
drives towards the root and takes the first available spot. If no spot is
found, then it exits the tree without parking. In this paper, we provide a
criterion to determine the phase of the parking process (subcritical, critical,
or supercritical) depending on the generating function of $ \mu$.
