We apply the boson exchange parquet solver for dual fermions to the half-filled Hubbard model on a square lattice at small interaction. Our results establish that, in this regime, nonlocal vertex corrections play an important role in the formation of the pseudogap. Namely, in comparison to the simpler ladder approximation, these additional vertex corrections included in the parquet equations enhance the coupling of spin fluctuations with the quasiparticles. The pseudogap thus opens already at a higher temperature, in quantitative agreement with the numerically exact diagrammatic Monte Carlo. The representation of the parquet diagrams in terms of boson exchange facilitates large lattice sizes and gives rise to an unbiased fluctuation diagnostic of the self-energy, which does not rely on the Fierz ambiguity. The fluctuation diagnostic implies that nodal and antinodal fermions are affected equally by spin fluctuations with the exact commensurate nesting vector (π,π). However, the antinode couples more efficiently to incommensurate fluctuations than the node, leading to the nodal/antinodal dichotomy. We corroborate this finding in terms of a spin-fermion-like calculation.