  We present an efficient implementation of the parquet formalism which
respects the asymptotic structure of the vertex functions at both single- and
two-particle levels in momentum- and frequency-space. We identify the
two-particle reducible vertex as the core function which is essential for the
construction of the other vertex functions. This observation stimulates us to
consider a two-level parameter-reduction for this function to simplify the
solution of the parquet equations. The resulting functions, which depend on
fewer arguments, are coined "kernel functions". With the use of the "kernel
functions", the open boundary of various vertex functions in the
Matsubara-frequency space can be faithfully satisfied. We justify our
implementation by accurately reproducing the dynamical mean-field theory
results from momentum-independent parquet calculations. The high-frequency
asymptotics of the single-particle self-energy and the two-particle vertex are
correctly reproduced, which turns out to be essential for the self-consistent
determination of the parquet solutions. The current implementation is also
feasible for the dynamical vertex approximation.
