Zeus in the Percy Jackson series is basically the ultimate plot device that makes everyone else's life harder. He's not around much, but when he is, it's always to drop a massive problem on everyone else's lap and then vanish back to Olympus. Think about 'The Titan's Curse'—his whole 'no kids of the Big Three' rule is what sets the entire modern plot in motion, forcing Thalia onto the scene and creating this ticking clock for Percy. His pride and paranoia are the engine for so much conflict.
His influence is indirect but enormous. He's the distant, angry dad whose bad mood means a storm is coming, literally and figuratively. In 'The Last Olympian', his refusal to help until the last second forces the demigods to stand alone, which is the whole point of the series. He's the obstacle, the unreasonable authority figure. The plot doesn't revolve around him; it contorts itself to avoid setting him off. Without that looming, petty, godly presence, half the tension just evaporates.