Parallel algebraic multigrid methods in gyrokinetic turbulence simulations are presented. Discretized equations of the elliptic operator −∇^{2}u+αu= f (with both α=0 and α≠0) are ubiquitous in magnetically confined fusion plasma applications. When α is equal to zero a "pure" Laplacian or Poisson equation results and when α is greater than zero a so called Helmholtz equation is produced. Taking a gyrokinetic turbulence simulation model as a testbed, we investigate the performance characteristics of basic classes of linear solvers (direct, one-level iterative, and multilevel iterative methods) on 2D unstructured finite element method (FEM) problems for both the Poisson and the Helmholtz equations.