Hmm, it's a tough one really. On the one hand you made an agreement, three years ago, that she should wait until 16. On the other hand, to stick to that just because it was said then, at a time when you didn't know what kind of person your dd would be like as she got older, seems a bit like sticking to a principle just for the sake of it. If the agreement had been that she should wait until she was 13, then there simply wouldn't be an issue now, would there?
You know what kind of person she is, pinkbubble. Does she still have a low pain threshold (complain about headaches, make a big fuss when she is ill etc)? Is she quite a mature 13 year old or is she easily led by her peers? What are her reasons for wanting them done now? What are her feelings about how it went three years ago?
IMO once a child becomes a teenager then they can earn the right to discuss things and possibly come to a compromise that suits both parties. It's all part of the process of growing up and learning to negotiate etc. If it was me, and I felt she had good reasons for wanting them done, and had thought about how she would feel about the pain side of it, then I think that her earning the money to pay for them to be pierced would be a good way to go about it.
A very long-winded way of saying what some others have already said !