Dontstopbelievin - a reverse is when the post is written from another person's point of view. So in this situation the OP is a teacher but wrote it from the point of view of the parent.
There is a misconception amongst some parents and students that teachers are not allowed to touch students. At one place I worked, if a male teacher touched a student for any reason, some students would yell out "PAEDO!" Horrific. The reality is that there is no law against teachers touching kids, and in fact Michael Gove, in one of his attempts to "give teachers more powers" enshrined in law that teachers could physically remove a disruptive student from class. I cannot ever imagine doing that. Can you? Can you imagine trying to physically remove a 15 year old from a room when he didn't want to go? It would be horrific. The teacher would look like an utter idiot as well, even if he/she was successful. It is so far from an "effective power" as to be utterly ridiculous.
Although it's allowed, experience tells you that you should never touch a child when you are cross or disciplining them, even if it is a very light touch, because it is very often construed as "an attack" and kids love to take revenge by claiming they were physically hurt. "You touched me" will always cause a teacher to panic because there are so many horror stories of kids then making false allegations against the teacher.
OP - did you actually talk to the parent and the parent demanded you apologise to their child for touching him? If so, what did you say? You need to report this to your line manager.
It wasn't clear from the first post if the students were meant to be coming into the room and sitting down or leaving the room? If they were meant to be coming in and were refusing, on another occasion tell them they have 30 seconds to be in their seats before you ask for a member of senior staff to remove them. Then walk away and let them choose. If they are meant to be leaving and won't get out the way of the door, do the same.
You have done nothing wrong by touching this child's shoulder - don't be harassed by them or their bullying parents if it is indeed an apology they are demanding.
Let me guess. Year 9?