In this paper, we study a parallel version of Galton-Watson processes for the random generation of tree-shaped structures. Random trees are useful in many situations (testing, binary search, simulation of physics phenomena,...) as attests more than 49000 citations on Google scholar. Using standard analytic combinatorics, we first give a theoretical, average-case study of the random process in order to evaluate how parallelism can be extracted from this process, and we deduce a parallel generation algorithm. Then we present how it can be implemented in a task-based parallel paradigm for shared memory (here, Intel Cilk). This implementation faces several challenges, among which efficient, thread-safe random bit generation, memory management and algorithmic modifications for small-grain parallelism. Finally, we evaluate the performance of our implementation and the impact of different choices and parameters. We obtain a significant efficiency improvement for the generation of big trees. We also conduct empirical and theoretical studies of the average behaviour of our algorithm.