I was in a very similar position with uni being the worst three years of my life and I didn't really have any friends on my course either.
I actually ended up doing a kind of 'middle ground' with graduating. I thought I'd be going, but my mum started a new type of chemotherapy a couple of weeks and was very very tired by then (she died just over 5 weeks later so was deteriorating at the time) before so I was worried about her having to sit through the day when she was feeling ill - although had I wanted to, she would have done it. But I was shy and anxious about the ceremony so I wasn't looking forward to it too much (and my parents had been to my brother's ceremony the week before and said sitting through watching everyone else go up gets a bit boring after a while!)
So we went up on the day, went and got the graduation robes, had photos taken in my robes etc and with my mum and dad, went out for lunch with my parents wearing my graduation robes, went to the cathedral and picked up the programme book so I could have a copy, took a couple more photos outside, then went and handed back the robes and went home.
So maybe there's something like that that you could consider if it would be easier - it might not be ideal, but it would mean your parents could have some photos of you 'graduating', because it sounds like they're very pleased for you either way, and it is nice in the future to have those photos at least.
If you're really, really unhappy about the graduation ceremony, speak to your parents and see what they think. But maybe consider if you can do something similar to the above if it would be easier than attending the actual ceremony.
Sorry if this isn't too helpful, but I hope work out because I can relate very much!