The nonlinear mechanism of the transformation of magnetohydrodynamic (MHD) Alfvén waves to kinetic Alfvén waves (KAWs) in the homogeneous magnetized plasma with small plasma parameter $\beta\,{\ll}\,1$ is investigated. As the generation mechanism, the parametric instability where the MHD Alfvén wave is the pumping wave is considered. On the basis of the two-fluid MHD and Vlasov equation, the nonlinear dispersion relation describing three-waves interaction, the instability growth rate and the threshold of the instability are found. The theoretical results are used for the interpretation of plasma heating in the solar corona.