To identify the joint parameters (e.g. the position of the joint center for a spherical joint, the position and the orientation of the joint axis for a revolute joint, etc.) from motion capture data, existing provably-correct algorithms require that at least three markers be attached to either of the two links adjacent to the joint. However, as shown in this article, it turns out that the identification of the joint parameters requires, for most types of joints, strictly less than three markers on any link. More precisely, we prove the structural identifiability of joint parameters in the following cases: (a) a spherical joint with two markers attached to each of the two adjacent links; (b) a revolute joint with two markers attached to one of the two links, and one marker attached to the other. We provide a practical algorithm to do the identification in case (a). Finally, we show that identification cannot be achieved with strictly fewer markers than listed in (a) and (b).