This work addresses the development and analysis of a second-order accurate finite volume scheme for parabolic equations with anisotropy on general simplicial meshes. The discretization involves only vertex unknowns without processing additional ones. The scheme construction makes use of a nonlinear transformation of the linear elliptic term. Two propositions are mainly presented for the approximation of the mobility function at the interfaces. The existence of positive solutions for the discrete system is guaranteed thanks to the proved a priori estimates. The energy dissipation of the scheme is moreover ensured. The convergence of the approach is established. Numerical tests are given to show the efficiency, accuracy and robustness of the proposed approach, with respect to the anisotropy, while a particular emphasis is set on the effects of the approximate mobility. They also confirm the obtained theoretical results, especially the decay of the free energy when time grows.