If you borrow another 20K you'd have nearly no equity, don't think the bank would go for that, unless the extension adds value.
Not a good idea to borrow from others if you were then refused the top up afterwards you wouldn't be able to pay it back.
Personal opinion of mine is that overdrafts are a very bad idea, particularly if they are long term, and even worse is going over the overdraft. You have savings so it was not necessary and now it means the bank will look more disfavourable on your application.
I've yet to see anyone do an extension that didn't go over budget. So budget that in.
If you fail with the bank, are you able to borrow the 20K from the parents and pay it off monthly? If they were agreeable to that of course. It's not a very large amount. Over say 5 years it would be about 330 a month.