L.,
I think you have the answer to your question better than anyone else can tell you here. You know your daughter and care for her more than anyone else. You also know that these days it isn't enough how much trust you have in your own child, there are many other things that fit into the equation that may be beyond your or your child's control.
It is difficult being a strict parent when everyone else's parents are much more lenient, but ultimately the decisions you make to protect your child will fall back on no one else but you. Therefore, trust your instincts and allow only that which gives you the greatest inner peace. Don't compromise your gut feelings or values for what others around you may be saying or doing. Remember, this is not about doing things in a socially aceptable manner or trying to gain anyone's approval, but doing that which will ultimately benefit your daughter the most.
You say your parents were overly strict, so you may want to look back and imagine what it would have been like if things had been different for you. Do you think you would have been better off, developed into a wiser more secure person? Or do you think perhaps your parents did the right things? At any rate, remember that things are quite different today than they were when most of us were teens.
Again, trust your deep instincts and make your decisions based on what your inner knower knows. Once you've made up your mind, let nothing or no one dissuade you, and take pride in knowing that you are trying to be the very best Mom your daughter can have.