As the other answer says, you will have difficulty getting accommodation unless one of you is over 18 and has a credit card.
However as you will be over 16, you can stay at a youth hostel, either sharing a dorm with strangers, or paying more to get a room just for your group. You will need to pay in advance, but if you are doing this several months in advance then you should be able to use a postal order if you can't get your parents to pay by card, and unlike a hotel, you don't need a card at check-in.
The SYHA have a hostel on Leith Walk, near the centre which is open all year:
http://www.syha.org.uk/hostels/central/edinburgh_central.aspx
A bed will cost £28 per night in August 2012
And another one which is only open in the summer (it is student accommodation):
http://www.syha.org.uk/hostels/central/edinburgh_metro.aspx
I would suggest booking by phone rather than online, and doing it fairly soon, preferably well before Christmas (they will already have bookings for next August).