  We present and implement a parquet approximation within the dual-fermion
formalism based on a partial bosonization of the dual vertex function which
substantially reduces the computational cost of the calculation. The method
relies on splitting the vertex exactly into single-boson exchange contributions
and a residual four-fermion vertex, which physically embody respectively
long-range and short-range spatial correlations. After recasting the parquet
equations in terms of the residual vertex, these are solved using the truncated
unity method of Eckhardt et al. [Phys. Rev. B 101, 155104 (2020)], which allows
for a rapid convergence with the number of form factors in different regimes.
While our numerical treatment of the parquet equations can be restricted to
only a few Matsubara frequencies, reminiscent of Astretsov et al. [Phys. Rev. B
101, 075109 (2020)], the one- and two-particle spectral information is fully
retained. In applications to the two-dimensional Hubbard model the method
agrees quantitatively with a stochastic summation of diagrams over a wide range
of parameters.
