Aiming at the problem that the vibration isolation performance of existing quasi-zero-stiffness isolators will be weakened under varying load conditions, an adjustable four-spring parallel quasi-zero-stiffness isolator (FPQZSI) is proposed by introducing the tilt linkage, which can actively adapt to load changes by adjusting the angle between the tilt linkage and the horizontal plane. The FPQZSI consists of a vertical spring that provides positive stiffness and three tilt springs, installed on the tilt linkages, that provide negative stiffness. According to the static equilibrium relationship, the conditions that the FPQZSI has zero stiffness at the equilibrium position are obtained. Then, the optimal structure parameters of the FPQZSI are determined based on maximizing the system quasi-zero-stiffness interval, and the quantitative relationship between the angles of the tilt linkage and the load mass is investigated. Furthermore, a unified dynamic equation for the FPQZSI under varying load conditions is established. The influences of the device structural parameters, the tilt linkage angle, and the load mass on the vibration response and displacement transmission characteristics of the FPQZSI are analyzed by using the harmonic balance method. The results indicate that by actively adjusting the tilt linkage angle, the designed FPQZSI can ensure good vibration isolation performance under varying load conditions.