In this work, a novel approach for versatile wrinkle-free application of (curved) pre-cut adhesive tapes on freeform 3D surfaces is presented. Straight and curved tape application paths are mapped onto the 3D object as geodesics and as lines with imposed geodesic curvature, respectively. The proposed surface-based path following control concept extends classical path following control by a novel parallel contact frame and a parallel projection operator. Using a static state feedback, the robotic system is transformed into a system with linear input-output behavior in the path coordinates. This allows to traverse a path on a 3D object with a draping roll without turning around the surface normal vector. The latter prevents distortions and wrinkles of the applied tape. Experimental results with a KUKA LBR iiwa 14 R820 demonstrate the feasibility of the proposed approach.