From how I see it, Gil at least knows when to back off when he's in a relationship. But when the relationship is broken up, he goes back to his inappropriate actions. Gil isn't the most "clingy" type so I don't see him going back to teasing Lars. It just doesn't feel right to tease your ex in front of people who know about your broken up relationship. Heck, even touching your ex is weird enough.
I can't touch on Gil's episode as I have never done it before, but if you take everything you said at face value, Gil had never touched Parker. Unless this conversation was past day 8, I don't recall them having this talk.
This all sounds new to me, so I'm going to interpret what you said. Even though I like Parker, he's the most normal to me, I'm not going to argue that he's not a flat character as I also think he doesn't bring much, yet. But I don't get what you mean by 'Gil asked Parker to stop him while dating Lars'. Are you saying that Gil told Parker to end their relationship?