  The parquet equations are a self-consistent set of equations for the
effective two-particle vertex of an interacting many-fermion system. The
application of these equations to bulk models is, however, demanding due to the
complex emergent momentum and frequency structure of the vertex. Here, we show
how a channel-decomposition by means of truncated unities, which was developed
in the context of the functional renormalization group to efficiently treat the
momentum dependence, can be transferred to the parquet equations. This leads to
a significantly reduced numerical effort scaling only linearly with the number
of discrete momenta. We apply this technique to the half-filled repulsive
Hubbard model on the square lattice and present approximate solutions for the
channel-projected vertices and the full reducible vertex.
