This article is devoted to studying dual regularization method as applied to parametric convex optimal control problem of controlled third boundary-value problem for parabolic equation with boundary control and with equality and inequality pointwise state constraints. These constraints are understood as ones in the Hilbert space $$L_2$$. A major advantage of the constraints of the original problem which are understood as ones in $$L_2$$ is that the resulting dual regularization algorithm is stable with respect to errors in the input data and leads to the construction of a minimizing approximate solution in the sense of J. Warga. Simultaneously, this dual algorithm yields the corresponding necessary and sufficient conditions for minimizing sequences, namely, the stable, with respect to perturbation of input data, sequential or, in other words, regularized Lagrange principle in nondifferential form and Pontryagin maximum principle for the original problem. Regardless of the fact that the stability or instability of the original optimal control problem, they stably generate a minimizing approximate solutions for it. For this reason, we can interpret these regularized Lagrange principle and Pontryagin maximum principle as tools for direct solving unstable optimal control problems and reducing to them unstable inverse problems.