My family owned and operated a private campground for many years. The biggest reason that they do not allow older units is because they don't want to get stuck with them. People will show up, pay the nightly fee, and then abandon their old trailer there. It happens more than you'd think. It's not just for snobby clientele. My parents have turned away some people/trailers because they looked too trashed. It's really expensive to scrap a trailer!
If you have an older unit and it's still very nice, you can tell them that over the phone. My parents definitely let older units into the park when the people over the phone said, "it's completely renovated, it's a well-taken care of classic, it's not junk, it's beautiful, etc".
Never lie about the length of your trailer! If you really want to piss private campground employees off, show up with a 44' 5th wheel when you reserved a 30' site on a holiday weekend. There's a reason they asked you the length. We only had a few sites that could fit 40'+ units.
If you're staying in a vehicle and want to stay at a private campground, It's just easier to ask for a tent site. Double check that you can park ON your site, though. Some private parks will have separate parking a little ways off. You can also ask for tent sites with or without electric and/or water. If you really need to stay at a private camp and it's sold out, ask if there's room by the barn, pavalion, staff area or even the dump station. Make it clear that you don't care where you stay. Be pleasant. They might find room for you someplace. They'll definitely be more inclined to find room for you if you're paying cash