I just accept the fact, I squeeze all the excess water out of the towel though, so it's not a big issue.

Cold showers didn't work for me, as you'll dry in like 15 minutes, and you are back to square one.

The damp towel and a light breeze can be amazing! Dries out in most environments too. Just flip the bed covers over and or set them aside and it all is set for the next evening, in my experience.

I think the parent might be suggesting that sleeping on a wet bed might trigger peeing reflex.

Eww. Never had an issue like that.

You could buy a mattress cover maybe?