Let your mum have her way with unpacking your stuff, this is a far more emotional day for her than it is for you!
You can just rearrange everything and make the room yours once they've gone.
I'd recommend not having your parents hang around though, if you couldn't bear to be apart from them for a long time, that's an issue which needs to be sorted before you arrive, so that you're not clinging on and being emotional on your first day. But you don't sound like that is going to be a problem. No need to rush them off though, you will probably have stuff to do, and if not then you will likely want to get to know the other people you're living with or near when you first arrive - someone new arriving usually prompts people to poke their heads out and say hello. I'd also advise not having your parents stay near for the night. They should take you down, help you move stuff in, say goodbye and leave. Having them there the next day will drag the whole thing out and make it too emotional, and you may well find you want to do something with people you're living with. (Jump on every oppertunity even if it sounds boring. It beats sitting in your room all day!)
When I moved in to Uni, I arrived reasonably early in the day. My dad and I unpacked things from the car and my mum started putting things away in cupboards and continued this while I chatted with other housemates and put things in the kitchen. Not only is it helpful, but you make your mum feel wanted if you allow her that. Like I say, you can always move things around after.
EDIT: With regards to mingling with flatmates -
Mingle as much as you can. Talk to everyone. Remember their names and try and remember what course they're doing. There's a load of 'standard' questions you can use to get a conversation going such as course, hometown, sports etc. If there's a few of you and nobody looks busy, suggest doing something, the Uni will likely have some introductory stuff on. It will be awkward, you don't know them and you don't know if you even like them, but give them a shot. I get on with my flatmates, but am not really friends with any of them. Despite that though, I'm still glad I spent some time in Freshers with them. It means we know each other a bit better and it sure beat sitting in my room on my own organising my books.
EDIT 2: Agreed, don't fall for the 'Freshers is the best week of your life' crap. It's not. You don't know people and you can't have that level of fun with people you just don't know. You should be outgoing throughout though, generally after Freshers week is over people form into their groups. I can pretty certainly state that those you spent time with in Freshers will always be at an aquaintance level with you, that is, if you're not friends. They'll always be civil and pleasant with you and you can rely on them as someone to to talk to. Those who you don't spend time with during Freshers, you probably just won't know at all during the year unless they turn out to be on your course. I mean it, spend time with as many people as you can!
Also, I didn't believe the whole 'Your closest friends/the people you'll live with in year 2 will be those you meet on your course' thing, but it's true. Honestly. If you have any kind of small group teaching, make friends with the people in there.