In this paper, a phase-field-based lattice Boltzmann (LB) model is proposed for axisymmetric multiphase flows. Modified equilibrium distribution functions and some source terms are properly added into the evolution equations such that multiphase flows in the axisymmetric coordinate system can be described. Different from previous axisymmetric LB multiphase models, the added source terms that arise from the axisymmetric effect contain no additional gradients, and therefore the present model is much simpler. Furthermore, through the Chapmann-Enskog analysis, the axisymmetric Chan-Hilliard equation and Navier-Stokes equations can be exactly derived from the present model. The model is also able to deal with flows with density contrast. A variety of numerical experiments, including planar and curve interfaces, an elongation field, a static droplet, a droplet oscillation, breakup of a liquid thread, and dripping of a liquid droplet under gravity, have been conducted to test the proposed model. It is found that the present model can capture accurate interface and the numerical results of multiphase flows also agree well with the analytical solutions and/or available experimental data.