Learn Danish: Mine forældre bor i byen. - My parents live in the city.

The word mine is the plural form of "my". Since forældre (parents) is a plural noun, you must use the plural possessive word mine. You would only use min or mit when talking about a single person or thing.

Does bor mean "live" as in being alive, or "live" as in having a home?

It means "live" in the sense of residing or having a home somewhere. If you want to talk about being alive, you would use a completely different Danish verb, which is at leve.

Where is the word for "the" in i byen?

In Danish, you usually do not use a separate word for "the". Instead, you add an ending directly to the noun. The basic word for city or town is by. Adding the suffix -en makes it byen, which translates to "the city".

Does byen refer specifically to a city, or can it mean town?

It can mean either. Danish uses the word by for both towns and cities. If a Danish speaker really needs to specify a large city, they might use the word storby, which literally translates to "big city".