The operation of Thick Gas Electron Multipliers (THGEMs) in Time Projection Chambers (TPCs) using various gas mixtures has applications in various nuclear and particle physics experiments. Of particular interest in low-energy nuclear physics is study of nuclear reactions involving Ne isotopes. These reactions can be studied using Ne-based gas mixtures at low pressures in TPCs used as active targets for radioactive beams. We report on the low-pressure operation of THGEMs in Ne + H2 gas mixtures. We show that, since the Ne+H2 forms a Penning pair, higher gains with THGEMs are achievable compared to pure neon gas. Moreover, H2 acts as a quench gas allowing for higher THGEMs voltages while producing minimal background in reactions such as fusion compared to carbon-based neon gas mixtures. Detailed electron transport and amplification simulations have been performed and they qualitatively agree with the increased gain H2 provides in the Ne:H2 mixture. The higher THGEM gains that are achieved with a Ne:H2 mixture will enhance the study of Ne-based reactions in active-target detectors that have high-granularity pad planes, leading to higher spatial resolution heavy-ion track data.