Park on Orme circle or Park drive and walk across the bridge. Also, you could park in the back of Amsterdam and walk up the hill, but it may be possible to get ticketed (I never have been, but believe there are signs).
You're right, there is no dog park parking lot. But there is parking closer to the dog park on the street by the private entrance. There's also paid parking at the visitors entrance. But for free, your best bet is to park on the street as close as you can.
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