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If you check the satellite view on google maps, you can see lots of normal cars in the parking lot, and the road is paved, so any car can get there (unless there is ice and snow on the road).
The rock is right at the parking lot. There is an elevator that leads up to the rock, but it is out of order and will not be repaired before Thanksgiving 2018. That means you have to walk up the stairs, which is not too bad, but obviously not possible with a stroller.
You can find more details on their website (https://www.chimneyrockpark.com/)