I don't understand the people in here who seem quite disgusted and shocked at the prospect of 16 year olds sleeping in the same bed. My ex and me would always share a bed when we were 16, which my mother had no problem with - then again, she's pretty open about sex which obviously going to be the reason why they don't want you sharing a bed. Honestly, as much as parents may sneer at the concept of sex, at the age of 16 I would say you're old enough to decide you're ready for it, it's legal, and hey.. it's sodding fun, so why the hell not? As for pregnancy, I would assume you're sensible enough to use some form of contraception, making the problem (almost) void.
I have no idea how you can convince your parents I'm afraid, but I very disagree with their opinion and questionable logic..