Should I carry a passport around everywhere I go in the UK?

I've never needed to show my passport in the UK once I've arrived, and that includes checking in at hotels, arriving to visit a business, renting a car and so on. I have an obvious Canadian accent and do obvious tourist things in addition to occasionally doing business things. (With grown children, I never have to prove I'm old enough for anything.) In theory I had to show my passport to get my train pass validated, but I don't remember them actually asking for it.

That is in stark contrast to the USA, where I quite often have to show it in fairly mundane circumstances. They'll ask for a drivers license, and if yours is "out of state" then they want something else. Every hotel checkin, every car rental, picking up my badge at a conference, even buying sudafed at a drugstore all needed my passport. So clearly this varies from country to country.

You shouldn't need your passport everywhere you go, and I wouldn't risk carrying it with you at all times.

However, if you plan to go anywhere where you need to be at least 18 years old, you will need some form of identity. And in the UK passports and driving licenses are the most commonly used. A doorman at a bar wouldn't accept a college identity. However you can apply for a UK identity card, such as CitizenCard, which means you don't need to carry your passport with you.

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.

Car rental asks for passports for non-UK residents as a surety for where to bill you if you fail to return the car etc. Same with most of the hotels and self-catering places we stayed at last summer. They want to see it, and some take photocopies of it for their records. If you live in the UK, they have other resources to find you.