The solution of a linear algebraic system with symmetric and positive definite matrix is considered for implementation of the preconditioned conjugate gradient method. Such a system arises as a discretization result of model elliptic partial differential equations. Two multilevel preconditioners are applied to avoid the heavy expenses of the direct solution of a linear system, namely the hierarchical basis (HB) preconditioner [cf. \textit{M. E. G. Tong}, Hierarchical basis preconditioners for second order elliptic problems in three dimensions, Ph.D. thesis, Univ. of Washington, Seattle (1989)], and the Bramble- Pasciak-Xu (BPX) multilevel preconditioner [\textit{cf. J. H. Bramble, J. E. Pasciak, J. Xu}, Math. Comput. 55, No. 191, 1-22 (1990; Zbl 0703.65076)]. The main idea is to choose the initial grid according to properties of the underlying operator coefficients. The parallel algorithm for BPX requires \(O(j)\) parallel steps for problems with \(j\) levels of the hierarchical basis, and the parallel algorithm for the HP preconditioner requires \(O([\log_ 2j])\) parallel steps. Numerical results on a Connection Machine are reported on.