Sun: can I live forever in a cave so I don't get any sunlight? No, you can't (in practical terms). But you can use sunscreen and you can limit the number of hours you get exposed to sun (don't be a dick and lay down on the beach 5h straight during hot summer days). Problem fixed.
Alcohol: can I live forever without drinking any alcohol. Yes, you can. Problem fixed.
Since you completely ignored the parent's valid point (many things that are in fact healthy for you can also cause cancer if overdoing, so the phrase "may cause cancer" by itself is not a valid argument), here's why your own argument has no merit whatsoever:
> Alcohol: can I live forever without drinking any alcohol. Yes, you can. Problem fixed.
I know the sun can increase skin cancer risk, that's why people take precautions.
The risks of alcohol are quite well known (e.g. what drinking & driving often does). It's also addictive, which means some people really struggle to stop or can't. It's not harmless at all.
There is so many behaviors and substances that somewhat increase cancer that we need some kind of scale to define what actually matters.
Personally a theoretical increase of 0.001% is kinda moot, even 1 or 2% would be something I'd see as a small risk given the sheer enjoyment I'd get out of alcohol (for instance, in comparison, moving to a greener city would probably decrease my overall cancer risk by much more)
But the sun, like alcohol, isn't so bad in moderation