I suppose it depends what you are defining as a serious relationship. Anything a fourteen year old does is incredibly serious as it reflects the intensity of their hormonal and developmental development. Gender doesn't make a difference, communication, trust and love makes all the difference.
I don't think as a parent that it is my place to "allow" or "not allow" such things. It sounds like a very restrictive attitude and not one that would allow a child to grow up as an independent decision maker. I think if you start "putting your foot down" and "not allowing" something, it takes away the responsibility and choice from the child and is most likely to be self-defeating anyway. Talking, discussing, giving them the tools to make the right decision are all far better options that will last them for a lifetime.
And no, gender makes no difference. Why would it?
If a 'full-blown' relationship includes sex, then no. If not, then perhaps. I think everybody is different and each situation is unique. If you forbid it, they are likely to push back. Been there and done that. I wish my parents had tried a different approach. Perhaps I wouldn't have wasted so much time trying to prove them wrong. Like Shelley, I believe communication, trust and love makes all the difference. Gender makes absolutely no difference.
No I wouldn't, and it would matter if they were a boy or a girl. They are not mature enough to be in that sort of relationship, and except in rare circumstances, do teen relationships last into adult life. You should save serious relationships until you are over 18.
I don't think 14 years is old enough to make a decision on relationships. However, if the emotions are strong and intense, then having a heart-to-heart discussion would be imperative. A blunt "no you are not allowed" would probably drive the child away from the parent. It's very important to gain a teenage child's trust.
There is the danger of having the child rebel and do it more anyway if you outright ban the relationship. Open communication is needed, and although you can voice your disapproval without being too harsh, you let them know that it's their decision, and explain if they think they are old enough for that type of relationship, then they are old enough for the consequences. I think it's important not to 'help' the relationship...eg...no sleepovers, not too much visiting or including them in family get togethers. Sooner or later, the relationship will fade out. I would hate it if they were having sex at that age, whatever their gender, but I think society would judge my daughter harshly but not my son. All the male relatives would be patting my son on the back.
The idea that we can allow or disallow our teens to be in relationships is pretty crazy. I wasn't allowed to have a boyfriend, but that never stopped me. I was never in a serious relationship, and that was probably worse in terms of emotional safety, I got hurt quite a few times, even though there was no sex.
When we ban teens from doing stuff they rebel, they're genetically programmed to rebel! So rather than giving my kids anything to rebel against I give them all the information they need to do it safely, and tell them I will love them no matter what. That way if they DO get into a relationship and they have trouble the know they can come to me. I couldn't go to my parents, I was lucky I had an older sister that I could trust.