We present a parallel quadtree algorithm that resolves between geometric objects, modeling space between objects rather than the objects themselves. Our quadtree has the property that no cell intersects more than one labeled object. A popular technique for discretizing space is to impose a uniform grid -- an approach that is easily parallelizable but often fails because object separation isn't known a priori or because the number of cells required to resolve closely spaced objects exceeds available memory. Previous parallel algorithms that are spatially adaptive, i.e., discretizing finely only where needed, either separate points only or make no guarantees of object separation. Our 2D algorithm is the first to construct an object-resolving discretization that is hierarchical (saving memory) yet with a fully parallel approach (saving time). We describe our algorithm, demonstrate experimental results, and discuss extension to 3D. Our results show significant improvement over the current state of the art.