Reconstructing the 3D articulated shape of an animal from a single in-the-wild image is a challenging task. We propose LEPARD, a learning-based framework that discovers semantically meaningful 3D parts and reconstructs 3D shapes in a part-based manner. This is advantageous as 3D parts are robust to pose variations due to articulations and their shape is typically simpler than the overall shape of the object. In our framework, the parts are explicitly represented as parameterized primitive surfaces with global and local deformations in 3D that deform to match the image evidence. We propose a kinematics-inspired optimization to guide each transformation of the primitive deformation given 2D evidence. Similar to recent approaches, LEPARD is only trained using off-the-shelf deep features from DINO and does not require any form of 2D or 3D annotations. Experiments on 3D animal shape reconstruction, demonstrate significant improvement over existing alternatives in terms of both the overall reconstruction performance as well as the ability to discover semantically meaningful and consistent parts.