Hey Tom,
Dogs are welcome. Dogs are everywhere, on a leash. Golden is very dog friendly. (If I thought I could contain my dogs in the Jeep, I would have mine with me too. Maybe we will try to bring our three legged fellow. If he gets away, I can usually chase him down.)
The Golden Rec Center is located 2-3 miles away from the cruise route. It is an easy in and out without having to back up. It is far enough off the beaten track that people less familiar with Golden wouldn't probably know about it. I am not positive about this, as I have never done this cruise before. There is so much parking back in that area. The Rec Center, the Library, the Golden Museum, the park, and a couple little league baseball fields. Most of the people partisipating in the cruise are parked in every nook and cranny on the parade route.
If you Google Map 'South Golden Road, Golden Colorado', You will see where the folks cruise and congregate. Compare that area to the Lyons Park area. I hope you can make it.
Tim