Gil must have touched Parker. In Parker Day7, when Walter surprised that Gil had never touched Parker, Parker's answer was very unnatural and explained that Gil's hormone secretion was strong, but it stopped four or five years ago. In Lars day4, when you choose to go to a restaurant to have dinner with Parker, Parker will complain to Walter about Gil's behavior of touching him.
In someone Day7, Walter told Gil that when Lars and Gil lived together, Parker wanted to see what Gil was doing. Gil said that Parker appeared to prevent him from making the same wrong decision, and then Lars and Gil broke up. This shows that Parker is involved in the relationship between Lars and Gil.
Yes, I also hope Parker has deeper reasons for doing these things. Instead of simply summing up the reason as Parker is a good man who has no brain to help others. No matter how good a friend is, he should not interfere in the affairs of couple. Otherwise, it will appear that Parker is very simple and flat character.
In fact, many players don't have a good impression of Parker, thinking that he is a wood, a serious, rigid, and a good old man who is slow to love.