Leashed pets are permitted in the park and preserve. However, we strongly encourage you to leave your pets at home, for your safety, their safety, and the health of the ecosystem. If your dog is already in Alaska with you, a quick Google search will give you a list of kennels that can board it for you during your trip to Lake Clark. Common park gateway communities with pet boarding facilities include Anchorage, Soldotna, Kenai, Kasilof, Homer, and Anchor Point.
Dogs running loose can bring enraged bears or moose back to their owners. They can also harass or kill local wildlife. While it's pretty obvious when a dog is chasing a moose or a squirrel, a loose dog can eat ground nesting bird eggs and chicks so quickly the owner never realizes it happened. They may also either contract a disease from wild canine populations, or introduce disease to fox, coyote, or wolf through their scat. Be aware that these wild canines are highly territorial, especially during summer denning season, and will kill loose dogs they encounter in their territory.
If you must bring your pet, remember that it must be leashed at ALL times, and that you must properly dispose of its scat by either burying it in a 6 inch cathole like you would your own waste, or by packing it out.