Accurate and nondestructive measurement of fruit ripeness is a critical requirement for intelligent agricultural systems, yet existing approaches rely heavily on visual cues and often fail to capture texture-related properties that are more directly linked to fruit quality. In this work, we present an instrumented parallel gripper with integrated tactile sensing for the quantitative assessment of peach ripeness. The system records force-displacement responses during controlled grasping, from which compressive stiffness is extracted as a tactile feature. To establish its physical relevance, the stiffness values were systematically compared with the elastic modulus obtained through three classical mechanical models, namely Hooke's, Hertz's, and Boussinesq's formulations. Experimental studies were conducted on two peach varieties ("Jinxiu" and "Baifeng") under different storage durations to capture progressive ripening effects. Results showed that compressive stiffness declined significantly with storage time for "Jinxiu" while remaining relatively stable for "Baifeng." Analysis of variance confirmed that fruit variety, storage duration, and measurement method all exerted statistically significant effects on elastic modulus (p < 0.001). Moreover, strong correlations were observed between compressive stiffness and elastic modulus, with Hooke's model yielding the closest agreement (r = 0.9234$ for "Jinxiu" and r = 0.8199 for "Baifeng"). These findings confirm that tactile stiffness features acquired by the parallel gripper provide a reliable proxy for elasticity-based ripeness characterization. Beyond demonstrating the feasibility of nondestructive ripeness measurement, this work highlights the potential of tactile-enabled robotic grippers to complement vision-based approaches in intelligent harvesting and postharvest quality monitoring, thereby advancing the development of practical and adaptive agricultural automation systems.