Home liability coverage does not generally cover any property other than the one that is insured. Under certain conditions, such as the home being next to a public park, your insurance may extend to the injury if your dog left the property chasing or threatening the injured person. On the other hand, if the dog was in a park with you and attacked someone, your homeowners insurance would not handle the claim.
One type of insurance coverage that would cover a dog attack in a public park would be personal liability insurance. This type of coverage is mobile, and protects you against injury or property damage claims anywhere you go. Instead of insuring your property the way car or home insurance does, personal liability coverage insures you and your actions directly related to you. Liability insurance can be included in pet insurance as well to protect the pet and injuries or damages caused by the pet.
In short, unless you can show that the dog attack is caused by something that happens on the insured property, homeowners insurance will not be much use in the claim. If you often take your dog to parks, shows, or along on trips, purchasing separate insurance is probably the most effective route to protect your financial interests.