They will absolutely let cars park in the lower lot and it's a short walk the campsites on the main circle. I just camped there a couple weeks ago and talked with them about this exact topic. The spots by the bath house are for the bathhouse not extra parking. Just make sure every vehicle gets a parking pass.
I asked the very same thing the family I was with was having to bring a third car. They had them Park in a lower lot down closer to the swim area and they were fine with it.
There's a few extra parking spots close to the bathhouse. There is room where the host is right as you enter the campground but you would have to run it by them.
Not sure about at the host station (there are several of them) but there is a lot for the day use area that you might be able to use. They're very nice so I'm sure the hosts would find you a place to park.
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