Every character falls into some classic love trope. In the nicest way, Gil is a playful man who falls into the "I'm not afraid to show my affection out loud" type of guy, which I'm guessing you know already.
If I'm not mistaken, Parker helped with the divorce papers, and Parker and Gil used to be a thing(?). (Parker was the first main storyline I did and don't remember much of it, correct me if I'm wrong. I started like a week ago :P.) With Gil's playful and touchy nature, it wouldn't be out far from the possibility that this is his way of thanking him?
I understand the argument with Parker said stop and Gil's not stopping, but maybe that's the reason for that idk.