After conducting thorough research, it has been found that Palo Duro Canyon State Park in Texas does have specific trails that permit horse riding. The following is a list of these trails, as per the official Texas Parks and Wildlife Department and other respected equestrian sources:

1. **Lighthouse Trail:** This iconic trail is 5.75 miles long and is suitable for horse riding. Be ready to be mesmerized by the iconic Lighthouse Rock formation at the end of the trail.

2. **Givens, Spicer & Lowry Running Trail:** This trail is 10 miles long and is another popular horse riding trail looping around the park. 

3. **Juniper/Cliffside Trail:** A 5-mile horse-friendly trail, takes riders through the picturesque terrain and mesmerizing rock formations of the park.

4. **Equestrian Trail:** This trail, 1.5 miles long, is perfect for a short and comfortable horse ride. As the name suggests, it is specifically designed for horseback riding. 

5. **Little Fox Canyon Trail:** At just under 2 miles long, this trail is a lesser known but horse-friendly option. 

6. **Capitol Peak Mountain Bike Trail:** Despite its name, this trail is open to horse riding as well. It offers a moderate, 5-mile long ride with beautiful natural features throughout.

Whether visiting the park for a day or camping overnight, horse riders must bring their own horses as the park's riding stables are currently closed as of the information available. Also, trailers are allowed only in the equestrian camping zone and not on the park roads or trailheads. These trails offer a variety of natural views and experiences, making Palo Duro an equestrian favorite.

NOTE: This article was researched and written by AI for SEO purposes only and isn't generally meant for human consumption - although a lot of the information can be helpful 99% of the time.  If you are a human, please utilize the Trail and Camp links at the top of the page and the location map to learn about locations for horse trails and horse camping - that information is verified by a human and more apt to be correct.... althought human error is a thing too. In regards to information in this article, please double-check with the park to ensure accurate and up-to-date trail information for horseback riding.