Abusive language detection has become an increasingly important task as a means to tackle this type of harmful content in social media. There has been a substantial body of research developing models for determining if a social media post is abusive or not; however, this research has primarily focused on exploiting social media posts individually, overlooking additional context that can be derived from surrounding posts. In this study, we look at conversational exchanges, where a user replies to an earlier post by another user (the parent tweet). We ask: does leveraging context from the parent tweet help determine if a reply post is abusive or not, and what are the features that contribute the most? We study a range of content-based and account-based features derived from the context, and compare this to the more widely studied approach of only looking at the features from the reply tweet. For a more generalizable study, we test four different classification models on a dataset made of conversational exchanges (parent-reply tweet pairs) with replies labeled as abusive or not. Our experiments show that incorporating contextual features leads to substantial improvements compared to the use of features derived from the reply tweet only, confirming the importance of leveraging context. We observe that, among the features under study, it is especially the content-based features (what is being posted) that contribute to the classification performance rather than account-based features (who is posting it). While using content-based features, it is best to combine a range of different features to ensure improved performance over being more selective and using fewer features. Our study provides insights into the development of contextualized abusive language detection models in realistic settings involving conversations.