I understand that this was likely a scary situation for you, your family, and possibly your puppies. Dogs pick up on our fear, so your puppies may have been responding to you as much as to a bigger dog running in their direction.
If the dog approached quickly, excited to play, but meant no harm and never acted in an aggressive manner then it would be difficult to prove any harm, or malicious intent. Most dogs can read another dog's intent via facial expression and body language. Running toward them with a happy, relaxed facial expression and bouncing around once the pup arrived and perhaps a play bow would signify an attempt to play.
Is this a park that many people bring their dogs to play in?
Were there other dogs there?
Were other dogs at the park off lead?
Given that there wasn't a sign that stated all dogs must be leashed this may be a case where there isn't any existing ordinance that requires dogs be leashed in public parks. If so and your pups were not physically attacked then although this was a scary incident there was no malicious or illegal intent.
That said even a friendly big dog needs to be under an owner's control. What if you and the pups were in a parking lot and the big dog ran to them to play and was hit?
I would contact your municipality and ask that signs be put up in the park if the local ordinance requires dogs to be leashed in public places. And for the safety of all, with or without a leash ordinance in your municipality, parks that allow dogs off-leash should be fenced if only to keep over-zealous dogs safe and to make clear where dogs can be safely off lead and where they need to be leashed.
I understand that you were frightened. And I too believe that the larger dog needed to be under much better control. Without seeing the event or knowing what the norms in that particular usually are however, or knowing your local ordinances, it's tough to say what should have been. I can only give you suggestions on how to avoid another such incident.