The obvious answer, of course, is that he's her father, so he's influenced her simply by living in the same household as her. However, the picture we get of him is not a particularly pleasant one. He's a man who enjoys drinking, tells rather outlandish stories of the past, likes to gamble, and believes women were meant to be servants to men. It's clear he treated his wife poorly, as Edna's husband assures us:
The Colonel was perhaps unaware that he had coerced his own wife into her grave
Edna, rather than seeing him as a blustering bully, now sees him as something amusing, something to watch and listen to and enjoy.
She had not much of anything to say to her father...; but he did not antagonize her. She discovered that he interested her, though she realized that he might not interest her long; and for the first time in her life she felt as if she were thoroughly acquainted with him. He kept her busy serving him and ministering to his wants. It amused her to do so.
Edna's father serves as an example in The Awakening of how Edna has come to view all men, even the one who believes women are lesser beings, meant to serve, and who must have treated her that way throughout her life. She has become independent and refuses to be subservient unless it suits her to do so.