The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M,L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change (L ̃^2 (x,y)=L^2 (x,y)+B^2 (x,y) with B∶= g(¯ξ(x),¯η); ¯ξ(x) being a parallel π-vector field) is established. The relation between the two Barthel connections Γ and Γ ̃, corresponding to this change, is found. This relation enables us to study the energy β-change of the fundamental linear connection of Finsler geometry: the Cartan, Berwald, Chern and Hashiguchi connections. Moreover, the change of their curvature tensors is concluded.