Rocky Mountain National Park is not especially pet-friendly, with dogs not allowed on any back country trails. As an alternative to hiking with your dog in the park, which you unfortunately cannot do, you might try hikes in the neighboring Indian Peaks Wilderness. Leashed dogs are allowed to hike on trails in the wilderness area as well as camp in the wilderness. There are many area hiking trails and wilderness areas to explore with your dog. Permits are required for those staying overnight in the area.