Always check with the park staff as it depends and can vary even among different campgrounds in the same NP. As far if I will ask or not depends on availability more than cost (most places I stay it's the same cost anyhow). Availability will also depend on the same time of the year. Julian Price on the Blue Ridge Parkway is a good example. Certain times of the year the tent section is fuller than the RV section, so then I stay in the RV section. At other times, RV has very few spots available but the tent section is wide open so then I will ask for a tent site with no problem to leave an RV site open for someone that really needs it.
One thing I should mention is my TC is just as quiet as a tenter with the exception of my waterpump. I don't have a generator, tv, or radio so no noise pollution except for the waterpump which I try to limit as much as possible when I'm in a tent site. That means I won't take a shower or wash dishes while there. I do this out of respect to the tenters that have also come for quiet. May not seem like a lot of noise for a quiet item, but when it's the only thing making noise, it's extremely loud to one trying to listen to the crickets.