We study the renormalization of the matrix elements of the twist-two non-singlet bilinear quark operators, contributing to the and moments of the structure functions, at next-to-next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the scheme and the regularization-invariant symmetric momentum subtraction (RI/SMOM, RI′/SMOM) schemes. The obtained results can be used to reduce errors in determinations of moments of structure functions from lattice QCD simulations. The results are given in Landau gauge.