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Understanding collective phenomena calls for tractable descriptions of correlations in assemblies of strongly interacting constituents. Capturing the essence of their self-consistency is central. The parquet theory admits a maximum level of self-consistency for strictly pairwise many-body correlations. While perturbatively based, the core of parquet and allied models is a set of strongly coupled nonlinear integral equations for all-order scattering; tightly constrained by crossing symmetry, they are nevertheless heuristic. Within a formalism due to Kraichnan, we present a Hamiltonian analysis of fermionic parquet's structure. The shape of its constitutive equations follows naturally from the resulting canonical description. We discuss the affinity between the derived conserving scattering amplitude and that of standard parquet. Whereas the Hamiltonian-derived model amplitude is microscopically conserving, it cannot preserve crossing symmetry. The parquet amplitude and its refinements preserve crossing symmetry, yet cannot safeguard conservation at any stage. Which amplitude should be used depends on physics rather than on theoretically ideal completeness.