Fast and safe motion is crucial for the successful deployment of physically interactive robots. Parallel robots (PRs) offer the potential for higher speeds while maintaining the same energy limits due to their low moving masses. However, they require methods for contact detection and reaction while avoiding singularities and self-collisions. We address this issue and present SafePR - a unified approach for the detection and localization, including the distinction between collision and clamping to perform a reaction that is safe for humans and feasible for PRs. Our approach uses information from the encoders and motor currents to estimate forces via a generalized-momentum observer. Neural networks and particle filters classify and localize the contacts. We introduce reactions with redundancy resolution to avoid type-II singularities and self-collisions. Our approach detected and terminated 72 real-world collision and clamping contacts with end-effector speeds of up to 1.5 m/s, each within 25-275 ms. The forces were below the thresholds from ISO/TS 15066. By using built-in sensors, SafePR enables safe interaction with already assembled PRs without the need for new hardware components.