Bilingual children at a young age can benefit from exposure to dual language, impacting their language and literacy development. Speech technology can aid in developing tools to accurately quantify children's exposure to multiple languages, thereby helping parents, teachers, and early-childhood practitioners to better support bilingual children. This study lays the foundation towards this goal using the Hoff Corpus containing naturalistic adult-child bilingual interactions collected at child ages 2½, 3, and 3½ years. Exploiting self-supervised learning features from XLSR-53 and HuBERT, we jointly predict the language (English/Spanish) and speaker (adult/child) in each utterance using a multi-task learning approach. Our experiments indicate that a trainable linear combination of embeddings across all Transformer layers of the SSL models is a stronger indicator for both tasks with more benefit to speaker classification. However, language classification for children remains challenging.