This area is referred to as an off-leash exercise area. There are wide open areas of fields that you can throw a frisbee, play fetch, whatever you would like. Plenty of space to really wear your canine out. It is rather unclear where the off-leash area is supposed to begin and end, but it is safe to assume that you cannot take your canine on the trails without a leash. This is due to the park being used for a variety of other recreational activities like biking, running, and hiking.
The Tarleton Park Off-Leash Dog Area requires that you do need to keep your canine in sight and under your voice control at all times. This is rather important to make sure that your canine is behaving and not bothering other people. It is also an unfenced area and you would not want your canine to end up lost in the vast trails of the park.
Restrooms are available at the entrance to the park, some shelters, and other facilities are available as well, but your canine is not allowed in those sections. Overall, this makes for an amazing place to come and kill a few hours with your canine friend. Enjoy a bit of sunshine and play some games with your canine friend to get out all that excess energy.