  We introduce new families of combinatorial objects whose enumeration computes
volumes of flow polytopes. These objects provide an interpretation, based on
parking functions, of Baldoni and Vergne's generalization of a volume formula
originally due to Lidskii. We recover known flow polytope volume formulas and
prove new volume formulas for flow polytopes that were seemingly
unapproachable. A highlight of our model is an elegant formula for the flow
polytope of a graph we call the caracol graph.
  As by-products of our work, we uncover a new triangle of numbers that
interpolates between Catalan numbers and the number of parking functions, we
prove the log-concavity of rows of this triangle along with other sequences
derived from volume computations, and we introduce a new Ehrhart-like
polynomial for flow polytope volume and conjecture product formulas for the
polytopes we consider.
