Tokamak fusion reactors are promising alternatives for future energy production. Gyrokinetic simulations are important tools to understand physical processes inside tokamaks and to improve the design of future plants. In gyrokinetic codes such as Gysela, these simulations involve at each time step the solution of a gyrokinetic Poisson equation defined on disk-like cross sections. The authors of [KKR21, KKR22] proposed to discretize a simplified differential equation using symmetric finite differences derived from the resulting energy functional and to use an implicitly extrapolated geometric multigrid scheme tailored to problems in curvilinear coordinates. In this article, we extend the discretization to a more realistic partial differential equation and demonstrate the optimal linear complexity of the proposed solver, in terms of computation and memory. We provide a general framework to analyze flops and memory usage of matrix-free approaches for stencil-based operators. Finally, we give an efficient matrix-free implementation fo the considered solver exploiting a task-based multithreaded parallelism which takes advantage of the disk-shaped geometry of the problem. We demonstrate the parallel efficiency for the solution of problems of size up to 50 million unknowns.