For b=(b1,⋯,bn)∈Z>0n, a b-parking function is defined to be a sequence (β1,⋯,βn) of positive integers whose nondecreasing rearrangement β1′≤β2′≤⋯≤βn′ satisfies βi′≤b1+⋯+bi. The b-parking-function polytope Xn(b) is the convex hull of all b-parking functions of length n in Rn. Geometric properties of Xn(b) were previously explored in the specific case where b=(a,b,b,⋯,b) and were shown to generalize those of the classical parking-function polytope. In this work, we study Xn(b) in full generality. We present a minimal inequality and vertex description for Xn(b), prove it is a generalized permutahedron, and study its h-polynomial. Furthermore, we investigate Xn(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.