Moved into girlfriend family home, whippet (3y, male) looking to me for approval over owners

So I moved into my girlfriends family home, saving for our own house. They had a whippet who I have known since he was a puppy. I was around often as he grew on weekends etc and would always play with him as I saw him on occasion.

Now after moving in I take him for walks on the weekend when he would be left for longer times alone due to routines and also I would often run with him, even sprint after him and play chasing.

However he is starting to validate his owners commands with me. At the start I would just give the same command as girlfriends parents but realised that his asserting my authority. So I opted to ignore him completely, however instead of caving to other commands he stands next to me. So I walk away.

But he is starting to cry when he knows I’m home but can’t be with me. I am starting to never let him initiate contact hoping I can still have a good bond but for him to know it’s only when I approach him but it’s becoming problematic.

I like him and I am training him to not be reactive to other dogs after getting the look command working for small stimulus hoping to grow to dogs at distance and eventually dogs up close. But I need him to stop looking at me as boss. I feel me taking him for runs and playing with him is over stepping? Any advice is welcome