Their Relationship
To Parker, Eliot is more like a big brother, as he is very protective of not only her but the whole Leverage team. He insults her a lot; but much of it sounds not unlike a brother would to his annoying little sister. While he is turned off by her personality, that does not affect his trust in her competence as a thief. Sometime before Season 2 he taught her some Mixed-Martial-Arts techniques which allows her to be more physical on the con. They work very well together when paired up. Parker and Eliot have come to realize they have darker pasts compared to the rest of the team, and they will do things the others wont.
Eliot treats the crazy Parker like a younger sister, in more ways than one. Eliot is often baffled by Parker's behavior, but at the same time is very protective of her, not letting anyone hurt his baby sister; a good example of this is in the future job where Eliot offered his services to tear off the psychic's arms. Sometime in Season 2, he helped train Parker in hand-to-hand mixed-martial-arts. He also loves to annoy her and make fun of her the way a playful elder brother would. He is someone Parker trusts to have her back as he never let anyone or anything hurt her.