Banksia Park in Bulleen is a dog-friendly park. There is a large circular track of compact fine gravel leading between the trees. There are seats within the area and a drinking fountain (for both humans and dogs).
A park for dogs and their owners. Credits: Parks Victoria website
How to get to Bulleen
There are bus services from the Melbourne CBD to Bulleen. Go to the bus terminus of Swanston and Lonsdale Streets and take Route 905 (towards The Pines Shopping Centre). It takes about half an hour to get to the Bulleen Terminus at Thompsons Road.