Vancouver and its suburbs and the other cities in the entire lower mainland is all a dog friendly area. Actually all of British Columbia is a "doggie" province. There are lots of parks and other places to walk dogs, including off-leash areas. There are too many to mention here so you will need to do your homework, look them up online and find the ones that are closest to your own location. Here is some information with a map for just the Vancouver city area, it doesn't include all the other parks in the suburb cities and other parts of the lower mainland:
Dog off-leash areas | City of Vancouver
Fleas and ticks shouldn't be much of a problem in the inner city parks. However, if you are going into parks with heavily wooded or wild areas where there is lots of wildlife then the chances of encountering ticks can increase, (and further out in the mountains it will be pretty much guaranteed). So you will need to follow tick protocol and take the usual tick and flea precautions before and after walking your dog in those areas, that includes doing a thorough brushing and body inspection of your dog afterwards. Don't forget to inspect yourself for ticks too.
Take note that in the lower mainland and throughout all of BC, encounters with bears and cougars are fairly common, so when walking in dense greenbelt and wilderness areas the risks of running into bears and cougars and other large wildlife also increases. This following dangerous incident with cougars happened just a couple of days ago in a greenbelt area in the city of Port Coquitlam:
2 young cougars shot and killed in PoCo park after dangerous encounters | CBC News
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