  Classical Gon\v{c}arov polynomials arose in numerical analysis as a basis for
the solutions of the Gon\v{c}arov interpolation problem. These polynomials
provide a natural algebraic tool in the enumerative theory of parking
functions. By replacing the differentiation operator with a delta operator and
using the theory of finite operator calculus, Lorentz, Tringali and Yan
introduced the sequence of generalized Gon\v{c}arov polynomials associated to a
pair $(\Delta, Z)$ of a delta operator $\Delta$ and an interpolation grid $Z$.
Generalized Gon\v{c}arov polynomials share many nice algebraic properties and
have a connection with the theories of binomial enumeration and order
statistics. In this paper we give a complete combinatorial interpretation for
any sequence of generalized Gon\v{c}arov polynomials. First, we show that they
can be realized as weight enumerators in partition lattices. Then, we give a
more concrete realization in exponential families and show that these
polynomials enumerate various enriched structures of vector parking functions.
