The most dominant object in our solar system outside of the sun is Jupiter.
What you’re asking depends greatly on the order of accuracy you’re looking for; for the lowest order of accuracy, you consider just the sun and the orbiting object, which isn’t all that bad of an approximation, but the next step that would actually have a fairly recognizable affect on orbital parameters is Jupiter. Jupiter has cleared certain orbits entirely in our system because of resonance, and often time perturbs the orbits of asteroids and other objects coming from outside the solar system in a noticeable (sometimes particularly noticeable) way.
Saturn is one of the planets closest to Jupiter, and so from reasoning alone that gravitational forces are distance dependent, I would assume that has a big part to play in why Saturn’s parameters differ the most.
So to answer your question, let’s consider something important: while Kepler’s third law isn’t perfect, it’s a good approximation and gives us an idea of how objects behave generally speaking. So if Jupiter disrupts an orbit, all parameters (save maybe angular ones to a good approximation like argument of the periapsis, etc) will be affected, according (roughly) to the relationships found in Kepler’s 3rd law. No individual parameter can be changed without changing the others (again excluding angular stuff)
Jupiter is the next biggest problem in getting orbital parameters down; no single parameter is more affected by errors, they’re all still connected in a quantitative sense as Kepler’s 3rd law predicts