Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization-based meshing approaches, where this directly translates into a shape-aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale-dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non-sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale-invariant control parameter to the user. It also allows us to consider non-local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.