The analysis of parent-child interactions is crucial for the understanding of early human development. Manual coding of interactions is a time-consuming task, which is a limitation in many projects. This becomes especially demanding if a frame-by-frame categorization of movement needs to be achieved. To overcome this, we present a computational approach for studying movement coupling in natural settings, which is a combination of a state-of-the-art automatic tracker, Tracking-Learning-Detection (TLD), and nonlinear time-series analysis, Cross-Recurrence Quantification Analysis (CRQA). We investigated the use of TLD to extract and automatically classify movement of each partner from 21 video recordings of interactions, where 5.5-month-old infants and mothers engaged in free play in laboratory settings. As a proof of concept, we focused on those face-to-face episodes, where the mother animated an object in front of the infant, in order to measure the coordination between the infants' head movement and the mothers' hand movement. We also tested the feasibility of using such movement data to study behavioral coupling between partners with CRQA. We demonstrate that movement can be extracted automatically from standard definition video recordings and used in subsequent CRQA to quantify the coupling between movement of the parent and the infant. Finally, we assess the quality of this coupling using an extension of CRQA called anisotropic CRQA and show asymmetric dynamics between the movement of the parent and the infant. When combined these methods allow automatic coding and classification of behaviors, which results in a more efficient manner of analyzing movements than manual coding.