A　novel　local　and　parallel　multigrid　method　is　proposed　in　this　study　for　solving　the　semilinear　Neumann　problem　with　nonlinear　boundary　condition.　Instead　of　solving　the　semilinear　Neumann　problem　directly　in　the　fine　finite　element　space,　we　transform　it　into　a　linear　boundary　value　problem　defined　in　each　level　of　a　multigrid　sequence　and　a　small-scale　semilinear　Neumann　problem　defined　in　a　low-dimensional　correction　subspace.　Furthermore,　the　linear　boundary　value　problem　can　be　efficiently　solved　using　local　and　parallel　methods.　The　proposed　process　derives　an　optimal　error　estimate　with　linear　computational　complexity.　Additionally,　compared　with　existing　multigrid　methods　for　semilinear　Neumann　problems　that　require　bounded　second　order　derivatives　of　nonlinear　terms,　ours　only　needs　bounded　first　order　derivatives.　A　rigorous　theoretical　analysis　is　proposed　in　this　paper,　which　differs　from　the　maturely　developed　theories　for　equations　with　Dirichlet　boundary　conditions.