Target-specific stance detection on social media, which aims at classifying a textual data instance such as a post or a comment into a stance class of a target issue, has become an emerging opinion mining paradigm of importance. An example application would be to overcome vaccine hesitancy in combating the coronavirus pandemic. However, existing stance detection strategies rely merely on the individual instances which cannot always capture the expressed stance of a given target. In response, we address a new task called conversational stance detection which is to infer the stance towards a given target (e.g., COVID-19 vaccination) when given a data instance and its corresponding conversation thread. To tackle the task, we first propose a benchmarking conversational stance detection (CSD) dataset with annotations of stances and the structures of conversation threads among the instances based on six major social media platforms in Hong Kong. To infer the desired stances from both data instances and conversation threads, we propose a model called Branch-BERT that incorporates contextual information in conversation threads. Extensive experiments on our CSD dataset show that our proposed model outperforms all the baseline models that do not make use of contextual information. Specifically, it improves the F1 score by 10.3% compared with the state-of-the-art method in the SemEval-2016 Task 6 competition. This shows the potential of incorporating rich contextual information on detecting target-specific stances on social media platforms and implies a more practical way to construct future stance detection tasks.