A locally free Lie group action on a closed manifold is called parameter rigid, if given another action with the same orbit foliation, there is a $C^\infty$ orbit-preserving diffeomorphism which conjugates one action to the other, up to an automorphism of the Lie group. We show some relationships between parameter rigidity and the first leafwise cohomology of the orbit foliation, and give a new example of a parameter rigid solvable group action. We also compute the first leafwise cohomology of the stable foliation of geodesic flows of closed orientable surfaces of constant negative curvature.