We consider parallel iterative processes in Krylov subspaces for solving symmetric positive definite systems of linear algebraic equations (SLAEs) with sparse ill-conditioned matrices arising under grid approximations of multidimensional initial-boundary value problems. Furthermore, we research the efficiency of the methods of moments for choosing an initial guess and constructing a projective-type preconditioner based on a known basis formed by the direction vectors. As a result, the reduction in the number of iterations implies an increase in their computational complexity, which is effectively minimized by parallelizing vector operations. The approaches under consideration are relevant for the multiple solution of SLAEs with the same matrices and different sequentially determined right-hand sides. Such systems arise in multilevel iterative algorithms, including additive domain decomposition and multigrid approaches. The efficiency of the suggested methods is demonstrated by the results of numerical experiments involving methodological examples.