We report the discovery of TOI-561, a galactic thick-disk star which has multiple planets amenable to atmospheric or surface characterization. This bright (V = 10.2) star hosts three small transiting planets identified in the NASA TESS Mission photometry: TOI-561 b (TOI-561.02, P=0.44 days, Rp=1.45 ± 0.11 R⊕), c (TOI-561.01, P=10.8 days, Rp= 2.90 ± 0.13 R⊕), and d (TOI-561.03, P=16.4 days, Rp =2.32±0.16 R⊕ ). The star is chemically ([Fe/H]= −0.41±0.05, [α/Fe]=+0.23±0.05) and kinematically consistent with the galactic thick disk population, making TOI-561 one of the oldest (10 ± 3 Gyr) and most metal-poor planetary systems discovered yet. We confirm the planetary nature of the candidates with high-contrast imaging, ground-based photometric follow-up and radial velocities from Keck/HIRES. Planet b has a mass and density of 3.2±0.8 M⊕ and 5.6±2.0 g cm−3, consistent with a rocky composition. Its lower-than-average density is consistent with an iron-poor composition, although an Earth-like iron-to-silicates ratio is not ruled out. Planet c is 6.5±2.4M⊕ and 1.45±0.55 g cm−3, and planet d is 2.4±2.0 M⊕ and 1.1±0.9 g cm−3, both of which are consistent with an interior rocky core overlaid with a low-mass volatile envelope. TOI-561 b is the first rocky world around a galactic thick-disk star confirmed with radial velocities and one of the best rocky planets for thermal emission studies.