Who gave the person their body? People don't create themselves, thus it is false to say that one's body belongs to them and they can do with it as they will. That is a lie. Instead, their body belongs to them as it was given to them for personal use and not to destroy and thus they also have the right over their body in comparison to others.
In my opinion our body does not belong to our parents, because then they could sexually abuse their child and enslave them, which is wrong. Rather, the creator of the world who gave you your life also gave you your body as a loan whilst you are in this world. Thus hurting yourself and ruining your body is forbidden, not only by others but by yourself too, you cannot decide to hurt your body. You must keep yourself healthy even when alone and none is looking up to you because the body is a loan to begin with and should be treated accordingly.
Thus each person has a responsibility to protect and safeguard their body, keep it healthy and not hurt it. The right is what to do with your body when talking about something that does not hurt it.
'Our bodies - to every hair and bit of skin - are received by us from our parents, and we must not presume to injure or wound them. This is the beginning of filial piety.' (Book of Filial Piety 1.1) When Xiahou Dun's eye was hit by an arrow, he immediately pulled it out. He swallowed it rather than throwing it away because he knew the eye belonged to his parents and he could not dispose of it. When Yue Fei was small, his father asked him if he would sacrificed his body for the country. Yue's replied that he would - if his father allowed him.
It is our duties to protect our bodies for our parent's sake. In fact, it's not only a filial duty: it's also a duty towards our children, siblings, cousins, friends, spouse, etc. You are particularly duty-bound to your children as you need to raise them with your own strength; by not protecting your body, you are depriving yourself of the ability to provide for them. Protecting your body is also saving your siblings, friends, etc., from worrying about you unnecessarily.
Thus it is simply not true that people 'own' their bodies and can do anything with them as long as they do not harm others. By taking recreational drugs with known side-effects on the body, you shirk your responsibility towards your parents, among others. The same argument goes for suicide (but not euthanasia).
For many individuals, it is a responsibility as a role model to set the good example. Throughout life, many will go from it being a right (optional) to a responsibility. Mothers and fathers have the responsibility to set a good example for their children, and body health and maintenance is part of that good example whether they realize it or not. Older siblings share that role model duty, as well as aunts, uncles, etc. Single men and women who know no one who idolizes them have no true necessity to set a good example for anyone, so it becomes an optional right. It just depends on how powerful your influence is upon other individuals.
Being a responsibility makes no sense. It's your body, it does not belong to anyone else, and it's completely up to you what you do with it. While yes, I can see a few exceptions, such as if you have kids who you need to raise (and in my eyes, are ethically bound to do so), everything else is down to you and you alone.