This paper describes a multistage framework for face image analysis in computer-aided speech diagnosis and therapy. Multimodal data processing frameworks have become a significant factor in supporting speech disorders’ treatment. Synchronous and asynchronous remote speech therapy approaches can use audio and video analysis of articulation to deliver robust indicators of disordered speech. Accurate segmentation of articulators in video frames is a vital step in this agenda. We use a dedicated data acquisition system to capture the stereovision stream during speech therapy examination in children. Our goal is to detect and accurately segment four objects in the mouth area (lips, teeth, tongue, and whole mouth) during relaxed speech and speech therapy exercises. Our database contains 17,913 frames from 76 preschool children. We apply a sequence of procedures employing artificial intelligence. For detection, we train the YOLOv6 (you only look once) model to catch each of the three objects under consideration. Then, we prepare the DeepLab v3+ segmentation model in a semi-supervised training mode. As preparation of reliable expert annotations is exhausting in video labeling, we first train the network using weak labels produced by initial segmentation based on the distance-regularized level set evolution over fuzzified images. Next, we fine-tune the model using a portion of manual ground-truth delineations. Each stage is thoroughly assessed using the independent test subset. The lips are detected almost perfectly (average precision and F1 score of 0.999), whereas the segmentation Dice index exceeds 0.83 in each articulator, with a top result of 0.95 in the whole mouth.