I can certainly relate to this situation.
The doc I saw thinks that there are strong signs I am somewhere on the spectrum, but I would need to see a specialist to get it confirmed whether it is AS or whatever, and full-on diagnosis would involve parental input.
Without knowing your parents, I can only say from the situation you have described that it sounds as if they simply don't want to change their views of you and your behaviour - something dx would mean they have to consider.
Mine are heavily invested in their views of me, and I think it would take a miracle to get them to view me otherwise: awkward and difficult, stubborn, cowardly, eccentric, unemotional, the list goes on. Also, they have convinced themselves that the reason I didn't do well at school was down to just plain lack of effort - I was bright, but struggled with things like getting on with the teachers and other students, time management and planning, remembering to do homework etc. After the school psychs had twice had me do every possible test but come up blank (there was no such thing as AS in the 1970's/80's) and refused to recommend me for additional help because I was off the ability scale in the wrong direction to get "statemented", this served only to cement in my parents' minds that I just didn't apply myself, and the levels of pressure and punishment escalated.
I feel almost certain they would conveniently "forget" the sort of details that would lend weight to a diagnosis, while blaming me for behaviours that they didn't know how to control - such as when something had really grossed or freaked me out, I'd be struggling to get away in terror, and in their minds the appropriate parental response to this was a hard smack.
If you decide to go for full dx, just do it anyway.