I used to be in the same boat as you (and also effectively high-functioning autistic) - even with my own PC, my mum was wanting to know what I was up to, in too much detail. In the end, I developed the ability to be doing two things at once - one that would satisfy my parents and what I was actually doing.
I also discovered the wonders of a multidesktop environment, with a key combo to switch desktops.
With gaming, my parents didn't mind me playing games, but my mum didn't really like the games I was playing, so I'd have to pause the game when she came in my room (she's not good with gorey movies, let alone games). That got annoying sometimes, if she had to come in my room a few times ><.
My mum did get really funny with MMOs though. She does sort of believe in the internet-is-full-of-predators bit. I've now played MMOs for a few years now, and nothing's happened to me, and in fact, had far worse things happen with people I met in RL than people I met online :/.
My mum wasn't very good at giving me space, and she didn't accept that sometimes I want to be alone. I remember arguing with my mum that I didn't want to eat breakfast with her, because I prefer to eat breakfast alone (and still do to this day). I don't mind talking after I eat, but I want eats first, preferably in peace and quiet. This was probably when I was 22 >< (I remember it being after uni, but before I moved out).