  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 $(\pi,\pi)$.
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.
