A lot of folks have been to this National Park and it always well worth a revisit if you have been. And if you have never been. You must take a Saturday out when staying at the cabin and take your family. The park is located roughly 14 miles from the Cabin just south of the Control road on your way to pine. The entrance is $5 per person and I believe they do not let dogs in, or charge extra for dogs. If you want to bring your critter to the park, please make sure you check with them first by calling. (928) 476-4202.
Here is a map running the control from the cabin. Its roughly 14.2 miles according to google. The control road is not a bad drive at all. I prefer taking it if I have a choice. I would suggest getting their when they open to avoid the crowds or if you would like to hike Pine trail. This trail runs through one side of the bridge up to the northern part of the park. It is well worth the effort though a little challenging when navigating through the cave on slippery rocks.
We have gone several times and enjoy having a nice picnic in the park above the cave under the oak trees at the North end. Seating does become very limited if you go later unless you get their early to claim benches. But take your time and you will find the park has waves of visitors coming and going. Here is the link to the main website with all the info and a map link to the park itself. Lots to see and plenty to enjoy! Thanks