  We exploit the parquet formalism to derive exact flow equations for the
two-particle-reducible four-point vertices, the self-energy, and typical
response functions, circumventing the reliance on higher-point vertices. This
includes a concise, algebraic derivation of the multiloop flow equations, which
have previously been obtained by diagrammatic considerations. Integrating the
multiloop flow for a given input of the totally irreducible vertex is
equivalent to solving the parquet equations with that input. Hence, one can
tune systems from solvable limits to complicated situations by variation of
one-particle parameters, staying at the fully self-consistent solution of the
parquet equations throughout the flow. Furthermore, we use the resulting
differential form of the Schwinger-Dyson equation for the self-energy to
demonstrate one-particle conservation of the parquet approximation and to
construct a conserving two-particle vertex via functional differentiation of
the parquet self-energy. Our analysis gives a unified picture of the various
many-body relations and exact renormalization group equations.
