North Vancouver is extremely accessible to Vancouver. And Vancouver is extremely accessible to the North Shore and its parks. You have the choice of using the Seabus (a fast quick ferry – apx. 20 min. or less) between Lonsdale Quay in North Vancouver and downtown Vancouver, or fast efficient busses from most anywhere on the North Shore to downtown.
It boils down to whether you’d prefer sleeping in a suburban or city setting.
Accommodations on the North Shore will be cheaper than equivalent accommodations in Vancouver , but you can fine a good price range in both. In Vancouver most downtown will charge extra for parking, in North Vancouver (or West Vancouver, which is on the North Shore too) most won’t charge for parking, though a few will. But you won’t need a car in either location if you prefer not.
You can easily get to most of the North Shore parks via bus. However, you’ll have to restrict yourselves to their very generous schedules and routes … not really a problem, though not quite as convenient as driving.
If you are into outdoor activities, I have a few suggestions:
Both Mt Seymour Park and Cypress Park are at the summit of the North Shore mountains. Both have lots of trails and hiking – a relatively tame wilderness experience within 30 minutes of downtown Vancouver. … and both have excellent views over the city (Cypress being better for views, Seymour better for hiking … but both are good for either). Treat both as if you were hiking in the wilderness miles from the city … people do get lost, but neither is intimidating if you hike a lot. Or one could take the gondola to the top of Grouse Mountain ski area for a more commercialized experience.
And in Vancouver, you could rent bikes and explore the seawall … a walking and biked route that follows the entire perimeter of the main part of Vancouver for many miles at the edge of the water. It starts downtown on the edge of Burrard Inlet (adjacent to the convention center), goes through (around) Stanley Park (wonderful park), along the downtown side of False Creek (beaches) to Yaletown area (one of the newer high-rise sections of Vancouver), thence around the east end of False Creel to its south side. There you could continue to Granville Island (one of Vancouver’s tourist areas), and from there either continue west as far as you’d want, or take the water taxi across False Creek back towards the city center.
On Granville Island (or other locations too) you could rent Kayaks if that's of interest.