Measurements of neutral current cross sections for deep inelastic scattering in e + p collisions at HERA with a longitudinally polarized positron beam are presented. The single-differential cross-sections dσ/dQ 2 , dσ/dx and dσ/dy and the reduced cross section σ ˜ are measured in the kinematic region Q 2 >185  GeV 2 and y<0.9 , where Q 2 is the four-momentum transfer squared, x the Bjorken scaling variable and y the inelasticity of the interaction. The measurements are performed separately for positively and negatively polarized positron beams. The measurements are based on an integrated luminosity of 135.5  pb −1 collected with the ZEUS detector in 2006 and 2007 at a center-of-mass energy of 318 GeV. The structure functions F ˜ 3 and F γZ 3 are determined by combining the e + p results presented in this paper with previously published e − p neutral current results. The asymmetry parameter A + is used to demonstrate the parity violation predicted in electroweak interactions. The measurements are well described by the predictions of the Standard Model.