  For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking
function is defined to be a sequence $(\beta_1,\dots,\beta_n)$ of positive
integers whose nondecreasing rearrangement $\beta'_1\leq \beta'_2\leq \cdots
\leq \beta'_n$ satisfies $\beta'_i\leq b_1+\cdots + b_i$. The
$\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the
convex hull of all $\mathbf{b}$-parking functions of length $n$ in
$\mathbb{R}^n$. Geometric properties of $\mathfrak{X}_n(\mathbf{b})$ were
previously explored in the specific case where $\mathbf{b}=(a,b,b,\dots,b)$ and
were shown to generalize those of the classical parking-function polytope. In
this work, we study $\mathfrak{X}_n(\mathbf{b})$ in full generality. We present
a minimal inequality and vertex description for $\mathfrak{X}_n(\mathbf{b})$,
prove it is a generalized permutahedron, and study its $h$-polynomial.
Furthermore, we investigate $\mathfrak{X}_n(\mathbf{b})$ through the
perspectives of building sets and polymatroids, allowing us to identify its
combinatorial types and obtain bounds on its combinatorial and circuit
diameters.
