Adding a damper in parallel with springs affects the overall stiffness by modifying the total impedance, which is represented as k + sc in the Laplace domain. The springs in parallel combine their stiffness according to the formula 1/k(total) = 1/k1 + 1/k2, while the damper introduces a viscous element that influences the system's response. The interaction between the spring constants and the damping coefficient alters the dynamic behavior of the system. Understanding this relationship is crucial for analyzing damped harmonic oscillators. The discussion highlights the importance of considering both spring and damper characteristics in system design.