I have a little different view. I think 13 is awfully young, but I've taught many a young girl who is dating and hiding it from her parents who say no. So, I would:
allow her to have the young man over to your house, with your supervision
allow her to double date with you and your spouse or you and a friend -- sit apart in the movie theater, but you are in control of the driving, etc.
allow for time at friends' houses with parents home -- call the parents and verify
talk to her honestly about sex and sexuality (even that it feels good and gets more and more tempting the longer you are with someone) and about your fears and desire for her
give her an age, or condition, when you will allow her to date 1-on-1
(Funny story, with my daughter, age 10, seeing me and Lebell dating, she has decided she won't have sex until she is 30 as it is just too grose. I assure her that if she decides when she is 28 that it is time, I will fully support her.
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