If you insure your own car in your mum or dad's name it could get awkward in the event of a claim. If you are the main driver they will not be amused if they find out.
Insurance companies are not completely dumb and if your mum/dad already have their own car(s) insured in their own name and you are the purchaser and registered keeper of the new car that you have also insured in their name - they are likely to smell a rat. Somewhere in the small print will lurk the sub-clause that gets them out of paying up if you have a bump.
If you get offered a good deal in your own name, start reading the policy from back to front - nasty stuff in contracts is never on the first couple of pages. Offering to pay the highest excess possible can get your quote down a bit - better that ?200 or so is sitting in your bank (and hopefully staying there) than part of your insurance payment and never being seen again.
If you are gong to save up for a bit, why not ring some companies and see if they will offer you a discount for pass plus and if so, how much. That way you can work out if it is worth doing or not.