This　paper　presents　a　new　type　of　local　and　parallel　multigrid　method　to　solve　semilinear　elliptic　equations.　The　proposed　method　does　not　directly　solve　the　semilinear　elliptic　equations　on　each　layer　of　the　multigrid　mesh　sequence,　but　transforms　the　semilinear　elliptic　equations　into　several　linear　elliptic　equations　on　the　multigrid　mesh　sequence　and　some　low-dimensional　semilinear　elliptic　equations　on　the　coarsest　mesh.　Furthermore,　the　local　and　parallel　strategy　is　used　to　solve　the　involved　linear　elliptic　equations.　Since　solving　large-scale　semilinear　elliptic　equations　in　fine　space,　which　can　be　fairly　time-consuming,　is　avoided,　the　proposed　local　and　parallel　multigrid　scheme　will　significantly　improve　the　solving　efficiency　for　the　semilinear　elliptic　equations.　Besides,　compared　with　the　existing　multigrid　methods　which　need　the　bounded　second　order　derivatives　of　the　nonlinear　term,　the　proposed　method　only　requires　the　Lipschitz　continuation　property　of　the　nonlinear　term.　We　make　a　rigorous　theoretical　analysis　of　the　presented　local　and　parallel　multigrid　scheme,　and　propose　some　numerical　experiments　to　support　the　theory.　(C)　2020　IMACS.　Published　by　Elsevier　B.V.　All　rights　reserved.