I believe your parent is alluding to the fact that if you are regularly active & get exercise, biking at an easy pace on flat land in good weather won't even cause you to break a sweat, whereas if you sit on the couch all day it will. (Hopefully don't need a source for this)
I would certainly like a source. As far as I can see, the only way to sweat less over time would be to lose weight. A more-conditioned cyclist still does the same amount of work and thus produces the same amount of heat.
A more-conditioned cyclist can perform the same amount of work with a lower heart rate & respiration rate. I don't have a source handy, but personally observe that my perspiration rate is directly related to my heart rate, and as I train & improve I can keep my jacket on performing the same work, where I needed to strip layers before.
I'm not a sports scientist but from what I read about and my experience with doing exercise, it modifies your metabolism so that the intensity of exercise that causes sweating goes up.
