Hi. I'm so sad reading your posts. Sounds like you are a great aunt. Can I say a couple of things, having been through this? Sounds like parents of child are embarrassed and afraid to confide in you. You are going to have to do this on your own- try to make a bond with this child despite them, not because of them.
My advice- notice what calms him down when he gets flapping and "stimming". Is is the fan? Is it running water? Maybe a flash light? Amybe make a straight line of Legos across the room and see what he does. If there is no effort from the parents to calm the child down with a dependable "thing", then you'll know why the child is having problems. All autistic children have an abnormal interest in something- you may need to help him find out what that is. Try running water when he is in the room. Does he come over and want to see it? Don't talk to him. Just have a stool nearby for him to stand on if he wants to. Turn on the lights and the ceiling fan a few times. Does he like it or get mad when it's off? Let him play with the switch by inviting him to play- non-verbal gestures are less threatening. Have a flash light or a hand held fan and play with it. If he wants it, he will have to interact with you to get it, even if he just moves a few feet closer to you. Maybe a jar of beans or some sand. Dig in the dirt in the front yard. Blow bubbles. Just keep trying. Play with it yourself- only one thing per visit. Get a small pop up tent and put it in your living room so he can have a quiet place from visiting relatives. Sit in it quietly. If he wants to come in, get out and let him have it.Get the picture?
Many of these children are PLACE specific, not people specific. My son, now 6, remembers people's houses where he was comfortable and found something that was stimulating to him from when he was just a baby. 100% of the time, he had a meltdown when leaving these places, and I did not understand it at the time, but it meant he wanted to stay and would rather be there in that "happy place".
Another thing- you somehow need to convince parents of the child to leave him with you or another trusted family member- but only after you have shown them you can make a "happy place" for the child. Children like this have no concept of time, so if you can keep him occupied playing in a running faucet for 4 hours, then let him play. These people need to get away and trust others more. It sounds like they cover up their anxiety about his behavior by overstating his abilities and underestimating his/their coping skills. All children, autistic or not, should be allowed- you got me- allowed- to develop relationships with other people. Did it ever occur to them that's why he is frustrated?
Just lower your expectations now for that "Auntie" relationship with hugs and games and know that someday very soon, with patience and love he will see you as his wonderful auntie, but in HIS way. And his behavior is not a reflection of their failures or yours, but you are speaking two different languages and it takes a while and an open mind to understand each other.
I used to have the attitude of your family, and I changed mine when a dear friend showed me what I was doing. It saved me from a hard life and frankly, it saved my son. He has wonderful adults in his life that know he'd rather have a trip to the top of the tallest building in town for his birthday, or a jar of old pennies or a set of bells for his Christmas present, things I never would have thought of but he treasures.
My son first smiled at a herd of goats at a petting zoo when he was 1. I dreaded the trip, but was forced to go by family. He laughed for the fifth babysitter (the other four got scared off by his constant screaming), uncontrollable, delighted laughter that he never did for me. I cried when I came home and heard him. Maybe your family hasn't had this breakthrough yet, but it can happen. You sound like just the girl to help.