Recently, neural network models which automatically infer syntactic structure from raw text have started to achieve promising results. However, earlier work on unsupervised parsing shows large performance differences between non-neural models trained on corpora in different languages, even for comparable amounts of data. With that in mind, we train instances of the PRPN architecture (Shen et al., 2018a)-one of these unsupervised neural network parsers-for Arabic, Chinese, English, and German. We find that (i) the model strongly outperforms trivial baselines and, thus, acquires at least some parsing ability for all languages; (ii) good hyperparameter values seem to be universal; (iii) how the model benefits from larger training set sizes depends on the corpus, with the model achieving the largest performance gains when increasing the number of sentences from 2,500 to 12,500 for English. In addition, we show that, by sharing parameters between the related languages German and English, we can improve the model's unsupervised parsing F1 score by up to 4% in the low-resource setting.