This work proposes a fully scalable parallel iterative solver for the time-harmonic Maxwell equations in a lossless medium with perfectly conducting boundaries and vanishing wave numbers. The authors adopt a mixed formulation of the Maxwell system, for which they apply the lowest-order Nedelec element space of the first family to approximate the electric field and the piecewise linear nodal finite element space to approximate the Lagrange multiplier. The discretization results in a linear saddle-point system. For the iterative solution of the linear saddle-point system, the authors apply the block diagonal preconditioning approach by C. Greif and D. Schötzau [Numer. Linear Algebra Appl. 14, No. 4, 281–297 (2007; Zbl 1199.78010)] as the outer iteration, and use the nodal auxiliary space preconditioning method by R. Hiptmair and J. Xu [SIAM J. Numer. Anal. 45, No. 6, 2483–2509 (2007; Zbl 1153.78006)] as the inner iteration for the shifted curlcurl operator. Algebraic multigrid technique is employed to solve the sequence of discrete elliptic problems involved in the inner iteration. For the Maxwell system with moderately varying coefficients, the preconditioned matrix is shown to be well conditioned, with its eigenvalues closely clustered. The performance of the proposal parallel solver is demonstrated on the Maxwell system with constant and variable coefficients. Extensive numerical results indicate a reasonable scalability of the solver with the mesh size on uniform, unstructured and locally refined meshes for the domains with complex geometries.