Using first-order smoothing theory, Fourier analysis and perturbation methods, we obtain the evolution equation of the wave spectrum as well as the nonlinear forces generated by Alfvénic turbulence in a finite-β plasma with dispersion and phenomenological Landau-damping effects. The stability of Alfvénic turbulence is then analysed by solving the derived mean-field equations. It is shown that parallel-propagating turbulent Alfvén waves with dispersion can be modulationally unstable, leading to amplification of large-scale Alfvén waves and acoustic waves when Landau-damping effects are strong and weak respectively.