We present high-order splitting time integrators for nonlinear evolution equations. Methods for splitting into two or three operators are introduced together with a rigorous error analysis and asymptotically correct defect-based error estimators. Our error analysis applies to equations of Schrödinger type, but also extends to parabolic problems within the appropriate functional analytic framework. The proposed methods are also demonstrated to be highly successful in a parallel environment, with good scaling in the number of processors, and to produce efficient and accurate simulations of intricate dynamics as in models for pattern generation.