  We establish a phase transition for the parking process on critical
Galton--Watson trees. In this model, a random number of cars with mean $m$ and
variance $\sigma^{2}$ arrive independently on the vertices of a critical
Galton--Watson tree with finite variance $\Sigma^{2}$ conditioned to be large.
The cars go down the tree towards the root and try to park on empty vertices as
soon as possible. We show a phase transition depending on $$ \Theta:= (1-m)^2-
\Sigma^2 (\sigma^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ \Theta>0,$
then all but (possibly) a few cars will manage to park, whereas if $\Theta<0$,
then a positive fraction of the cars will not find a spot and exit the tree
through the root. This confirms a conjecture of Goldschmidt and Przykucki.
