It's a specified dog beach rather than a park but Waterman's is a wonderful place for the dogs to swim and socialise. There is a natural pool formation in one part of the reef that many of the older dogs love to wade into. Great people to talk to as well.
I think one of the best dog parks in Perth would have to be Bayswater Riverside Gardens. There's wide open spaces for them to run around on, and they can also splash around in the Swan River. It's also a great spot for their human companions to chill out...plenty of picnic tables and seating, and a great playground for the kids.
I go also along the River near Bassendean/Ashfield.
It is a place called Sandy Cove and you walk right round the river, and heaps of space for the dogs to run, and also the river to swim in.
Very friendly humans also!