Phoneme-level computer-assisted pronunciation training systems typically rely on phoneme-level annotations, which are costly and scarce. In this work, we investigate whether phoneme-level mispronunciation information can be learned without phoneme-level supervision by exploiting higher-level pronunciation labels. Specifically, we study a weakly supervised setting in which models are trained using only utterance- or word-level pronunciation labels and analyze whether this supervision induces useful phoneme-level score predictions. We further consider a two-stage training scenario in which a model trained only with utterance-level labels is finetuned using a limited number of carefully-selected phoneme-level labeled utterances. We find that, using our proposed architecture and selection process, the two-stage process leads to comparable results to those obtained with full phoneme-level supervision, requiring only a small fraction of phoneme-level labels.