Dogs are allowed in some regional parks, but not in others. It’s true that dogs are not allowed on the Grouse Grind. And in 2018, dogs were banned on the adjacent BC Mountaineering Club trail as well, to reduce impacts and increase visitor safety.
The rules vary from park to park, and region to region. For example, Metro Vancouver has a Canine Code of Conduct within its regional parks, while the Capital Regional District has specific dogs-on-leash areas and seasonal prohibition zones. The best thing to do is look up the rules for the park you’d like to go to.