Degrees of freedom for a bead on a parabolic wire?

It has two degrees of freedom: its position along the wire and its rotation around the wire. But in most cases there is no interaction between the rotation and the position along the wire, so the equations of motion can treat the two degrees of freedom separately. That is, the equation describing rotation of the bead around the wire will not have any terms that include position along the wire, and vise versa. This means that for calculating motion along the wire, there is effectively only one degree of freedom.