In multilingual neural machine translation, it has been shown that sharing a single translation model between multiple languages can achieve competitive performance, sometimes even leading to performance gains over bilingually trained models. However, these improvements are not uniform; often multilingual parameter sharing results in a decrease in accuracy due to translation models not being able to accommodate different languages in their limited parameter space. In this work, we examine parameter sharing techniques that strike a happy medium between full sharing and individual training, specifically focusing on the self-attentional Transformer model. We find that the full parameter sharing approach leads to increases in BLEU scores mainly when the target languages are from a similar language family. However, even in the case where target languages are from different families where full parameter sharing leads to a noticeable drop in BLEU scores, our proposed methods for partial sharing of parameters can lead to substantial improvements in translation accuracy.