Two-Phase Thermal Management Systems (TPTMS) such as heat pipes and vapor chambers offer high heat flux thermal management for various applications in electronics, energy, high power battery, and spacecraft systems. The maximum cooling power of a TPTMS is limited by the coolant chocking within the wick structure, i.e., poor wickability via a viscous-capillary limit. To address this challenge, enhanced wickability is examined using non-uniform pore size wicks. A two-phase single component free-energy-based Lattice Boltzmann Method (LBM) is employed to numerically study the enhanced wickability by examining the pore-scale rate-of-rise through particle-filled parallel plates. The non-uniform pore size wicks are designed by zig-zag arrangement of the single/multiple arrays of uniform/non-uniform square particles between two parallel plates. The results show that the cumulative liquid saturation of the non-uniform pore size wicks (measure of permeability) increases up to 112% due to the presence of the large pores, while the capillary pressure increases by up to 145% from the presence of small pores, compared to the uniform pore size wick. The study also shows that the non-uniform pore-network within the bi-particle-size wicks (BPSW) improves the cumulative liquid saturation of the single- and three-columnar BPSWs by up to 53 and 18%, respectively, while their capillary pressure increases by up to 76 and 39%, respectively, compared to the Uniform-Particle-Size Wicks (UPSW). The study shows that increased pore-size ratio and small/large pore locality further enhances the cumulative liquid saturation of the single- and multi-columnar BPSW by 13 and 26%, while it shows the additional 38 and 30% increase in the capillary pressures, respectively. Also, graded wicks increase the capillary pressure due to the smaller pores at the top of the wicks, while they marginally decrease the rate-of-rise compared to the non-uniform pore size wicks. The study also examines the effects of porosity and particle distribution on enhanced wickability.