Well exactly. So how on earth you think the parents should be prosecuted is beyond me.
You've posted yourself about how few residential places there are in the UK.
What, exactly, would you have had his parents do? Chain him up so he can't leave the house? Bearing in mind his parents already had to bribe him to do things like take showers, do you really think they'd have come to no harm if they'd tried to stop him?
As you say, what exactly would you have social services, doctors, and foster carers do? Why do you think his parents must have had some magic solution where the professionals don't?
AR was known to the police. After they found him on a bus with a knife he was returned to his parents with no further action taken. He was known to Prevent, which did nothing despite multiple referrals.
He was known to multiple agencies.
Even if his parents had called the police that day when he got in the taxi, what exactly do you think would have happened, bearing in mind he was not under any high-risk flags and hadn't actually committed any crime at that point - how fast do you think the police would have responded? Fast enough to find where he'd gone, identify the taxi he got into, and get to the dance class before he made his attack? How realistic do you really think it is that the police would have got there on time?
The only person responsible for what happened that day is AR.
The failings of our social care system and mental health support is the responsibility of multiple successive governments.