Pets are allowed in many of the public parks in the Black Hills [1] area, but the rules are sometimes very restrictive. In the Badlands [2], for instance, pets are allowed only in developed areas of the park (that is, on roads and in campgrounds). Pets must always be on a leash and cannot go on hiking trails or enter any of the wilderness areas of the park.
At Mount Rushmore [3], pets, with the exception of service animals, are not allowed anywhere in the memorial. The Black Hills National Forest sites allow pets on the trails but they must be leashed or under strict voice control. There is an additional $2 charge for animals in campgrounds.
Custer State Park [4] allows pets in campgrounds, but pets are not allowed in any of the camping cabins, lodges, or other park buildings.
You are allowed to leave your pet in an unmanned vehicle, but this could be very dangerous. Summer temperatures can soar to over 100°F, and even on an 80°F day, the interior of a car can reach temperatures far above those safe for your pet. There are many pet-friendly lodgings in the Black Hills.
On a day when you expect to spend a great deal of time at a site that restricts pets, confer with your host to see if you might want to leave your animal for the day, or with a local vet or kennel.
Links:
[1] http://www.moon.com/destinations/mt-rushmore-the-black-hills/discover-mount-rushmore-the-black-hills
[2] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-badlands/badlands-national-park
[3] http://www.moon.com/destinations/mt-rushmore-the-black-hills/mount-rushmore-and-the-central-hills/mount-rushmore-national-monument
[4] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-southern-hills/custer-state-park