Generally speaking, dogs are allowed in the National Parks on the paved roads (leashed of course). They are not allowed off-pavement, backcountry. This isn't a perfect guide as there are a few dirt roads (like the road to the parking lot at the trailhead for Elephant Hill) that dogs are allowed on, but AFAIK dogs are allowed on all paved road areas.
But, as Sue has mentioned, most of the trails are not in the National Parks.