The most enduring mysteries in volcanology cannot be tackled by one method alone. Different methods give complementary insights on the same structures and processes. In this dissertation I use InSAR (Interferometric Synthetic Aperture Radar), gravimetry, and rock physics in concert with other data to investigate volcano-tectonic interactions and volcanic structure and life cycles. A major theme of my work is integrating geodetic methods with complementary data to synthesize a self-consistent picture of the volcanic system, including using rock physics to quantitatively link geology to geophysical parameters. The Central Volcanic Zone of the Andes contains informative end-members for studying the extremes of volcanic processes. Sabancaya volcano in Peru experienced some of the strongest earthquakes ever recorded near a volcano without coincident eruption, and Uturuncu volcano in Bolivia exhibits signs of unrest despite having no eruptions in over 250,000 years. At Sabancaya volcano I calculate a 4 year long InSAR time series, combining this with other InSAR time series and seismic, thermal, and geological data to determine that magmatic intrusion in fluid-saturated crust triggered the unusually strong seismicity at this volcano. I then turn to Uturuncu volcano, where I generate an updated density contrast model of the volcano's shallow subsurface. Combining this model with a published resistivity model, I argue that Uturuncu's unrest may be partially due to ongoing ore body formation. I build on this work with a quantitative, rock-physics based joint analysis of density contrast, resistivity, and seismic tomography models of Uturuncu's shallow subsurface. My self-consistent, petrologically informed analysis shows that shallow (<10 km) geophysical anomalies at Uturuncu are quantitatively consistent with a hydrothermal system with up to 8 vol.% of 3 wt.% NaCl brines and inconsistent with large zones of dacite melt or mush. At both Sabancaya and Uturuncu, my work demonstrates the key role of aqueous fluids in driving volcanic unrest and breaks new ground in synthesizing multiparameter data in the quest to understand the structure and workings of volcanic systems.