The quality of interactions between parents and children is a critical factor in child development. Recent years have seen programs to improve parenting behaviors through evidence-based approaches, such as attachment-based interventions. A vital element of these programs is to assess the quality of parenting behaviors via video recordings of parent-child interactions, which is often time-intensive. In our previous work, we explored machine learning models to predict expert ratings of parenting behaviors from video recordings of semi-structured parent-child play. However, the large set of low-level multimodal features struggled to provide explainable insights, which created barriers to communicating with domain experts and improving the models further. In this work, we developed a machine learning pipeline that combines sparse multiple canonical correlation analysis with causal discovery techniques to uncover explainable causal relationships between nine categories of behavioral features and the quality ratings of parent-child interactions. This approach offers valuable insights into the otherwise black-box models and contributes to the growing body of work on transparent and trustworthy machine learning models of parenting behaviors.