  At each site of a supercritical Galton-Watson tree place a parking spot which
can accommodate one car. Initially, an independent and identically distributed
number of cars arrive at each vertex. Cars proceed towards the root in discrete
time and park in the first available spot they come to. Let $X$ be the total
number of cars that arrive to the root. Goldschmidt and Przykucki proved that
$X$ undergoes a phase transition from being finite to infinite almost surely as
the mean number of cars arriving to each vertex increases. We show that $EX$ is
finite at the critical threshold, describe its growth rate above criticality,
and prove that it increases as the initial car arrival distribution becomes
less concentrated. For the canonical case that either 0 or 2 cars arrive at
each vertex of a $d$-ary tree, we give improved bounds on the critical
threshold and show that $P(X = 0)$ is discontinuous.
