The concept of parametric controllability as applied to systems of rigid bodies is discussed. The topic of discussion is Langrangian systems for which "unfreezing" of the parameters is possible such as, for example, the refinement of a model by taking account of the small variability of the links assumed by rigid bodies in the first approximation. As has been shown, just taking account of a small change in the parameters can ensure the controllability of a mechanism which was not controllable assuming rigidity absolute of the links. Certain sufficient conditions are proposed for parametric controllability in invariant manifolds for objects with cyclic coordinates. A two-link pendulum in the horizontal plane under the action of an internal moment (from the first link to the second) is considered as an example. The effect of its mass inertial parameters on the controllability is investigated. The parametric controllability of such an object in a manifold of zero angular momentum, due to the elastic pliability of the second link or the oscillations of an additional mass on a spring along the second link, is demonstrated. An example of a parametrically controllable planetary mechanism with slippage of discs is also considered.