A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial-boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.