There is still no legal requirement in the UK to carry your passport with you all the time.
Traveling around I take it with me, as it is safer on my person than left behind in a hostel or even in a hotel room.
But when staying with friends I leave it in their house.
I do usually also carry my national (EU) ID card but have not needed that any time in the UK, apart from when checking in at hotels.
In your case I would leave it in the place you live. Your parents may well want it with them when they travel around or stay in a hotel which they do not trust, but if they can leave it in a safe place where you live it is acceptable by law.
When this question was asked not many people would ask for your age when going into a bar or buying alcohol, these days that is more the norm. So if you live in the UK for longer, are or look like you are under 30, best get a recognized ID with your age on it. Your passport, or in many cases a national ID card from home, should work, but as not everybody in the UK knows those, you may want to get a UK based age card or driving license.