With ever-improving driver assistance systems and large touchscreens becoming the main in-vehicle interface, drivers are more tempted than ever to engage in distracting non-driving-related tasks. However, little research exists on how driving automation affects drivers' self-regulation when interacting with center stack touchscreens. To investigate this, we employ multilevel models on a real-world driving dataset consisting of 10,139 sequences. Our results show significant differences in drivers' interaction and glance behavior in response to varying levels of driving automation, vehicle speed, and road curvature. During partially automated driving, drivers are not only more likely to engage in secondary touchscreen tasks, but their mean glance duration toward the touchscreen also increases by 12% (Level 1) and 20% (Level 2) compared to manual driving. We further show that the effect of driving automation on drivers' self-regulation is larger than that of vehicle speed and road curvature. The derived knowledge can facilitate the safety evaluation of infotainment systems and the development of context-aware driver monitoring systems.