I was just speaking with a colleague about this earlier today. It's a very tricky situation.
On one hand, I firmly believe that a 13 year old young woman should be able to obtain contraception without necessarily gaining consent of their parents. The health care practitioner must follow agreed guidelines to ensure the young person is capable of giving consent. I also believe contraceptive services need to be as accessible as possible for young people because at the moment, they can be difficult enough to get to even if you are an adult. Many young people feel intimidated about going to the family doctor or two a mainstream clinic - perhaps fearing (sometimes rightly) that they'll be judged harshly.
It is my understanding that most sexual health and healthy respect programmes set up for young people DO push condoms, both to prevent pregnancies and STIs but also recommend other forms of contraception (so long as these are suitable for the individual) both as a back up and in the realistic knowledge that condom usage can be erratic amongst young people - and frankly, people of all ages.
So if a young woman is having sex or is likely to have sex at least having access to reliable contraception will reduce the risk of unplanned pregnancy - the impact of which can be devastating physically, emotionally, socially, educationally and economically for a young woman.
But, there IS a problem with this. Evidence is increasingly emerging to indicate that young women are being pressured, coerced or actually forced to have sex not just with a "steady boyfriend," but often with multiple partners - like a boyfriend wanting her to "do favours" for his mates. This is happening with girls at younger and younger ages (12 or 13 isn't unusual, in my experience.)
So, while a young woman may be mature enough to understand the implications of and give consent to be prescribed contraception, she may be in a situation where she actually doesn't have genuine control over when, how and with whom she has sex.
In such circumstances, condoms are unlikely to be effective (i.e. if she doesn't feel she can say no to sex, it's not likely she'll feel able to insist on condom use either.) Injections or implants will reduce the chance she'll get pregnant, so that's something at least. But something just doesn't feel right about this scenario. Whether the health care practitioner does or doesn't offer contraception, the young woman will still be pressured/coerced/forced to have sex and be at risk of sexual exploitation.
If the "boyfriend" or other men involved are over 16, it's a child protection issue, but that's rarely picked up on because it can seem heavy handed to report the situ to the authorities where it's just a "slightly older boyfriend" and "consensual" sex. But "older boyfriends," can mean something else and "consensual sex" is nonsense where genuine informed consent isn't involved.
It just feels like we're seriously failing a whole generation of young women if we seem only concerned about keeping them from getting pregnant (yes, that is important for their sakes as well,) but not that bothered if they have little control over their own bodies or when, where, how and with whom they have sex.