Lower Squamish River and Mamquam River have lots of room, backcasting room that is. It can be congested with anglers this time of the year, especially on weekends. Currently, there is a catch and release pink salmon fishery. If bringing home a fish is not an expectation and one just likes to practice casting and hook a few fish, then this maybe the place to go. The pink salmon fishing should be good for the rest of this month, while the chum salmon fishery will start around early October until early December.
Ambleside Beach/Capilano River mouth also has lots of room to flyfish, the coho salmon are just quite picky.
Garry Point Park in Steveston is also good for flyfishing at times, especially when the pink salmon start moving in, around late August. Just be very careful as there are many other recreational users in that park, so watch behind you when you cast and be nice to others so the park will remain anglers-friendly.
If you head towards the Fraser Valley, there are also many small streams where you can flyfish. Chilliwack River, Harrison River, Stave River are the major Fraser River tributaries where flyfishing is popular. With a bit of walking, you'll find plenty of room.