The direct route would pass through 2 national parks and 1 provincial park, the total number you'll go through will depend on what detours you make. But it doesn't really matter how many parks, just now many days. That is to say, a national park permit is valid for any national park, so it doesn't matter whether you're in one or three national parks for a given period of time, you just need a national park permit for that many days. Provincial parks in Alberta and BC don't cost any money or require any permits to visit, unless you're camping there.
If you are going to be in national parks for 7 days or more, it's most cost-effective to buy a one-year national parks pass, called the Discovery Pass. So, figure out how many of your 10 days will be in national parks (eg Banff, Jasper, Yoho, Kootenay). If you are going to be adjusting plans during your trip and you're not sure the answer, you can actually buy a national park permit for a few days and then upgrade to a Discovery Pass, with the cost of the permits deducted.
You will be passing through both national parks (for which you need to pay an entry fee) and provincial parks, for which there are no fees. You'll need to pay for a national park pass for Banff and Jasper National Parks, but since you are going to Vancouver via highways #16 and #5 instead of #1, you won't be passing through Glacier or Mt. Revelstoke National Parks, so you will only need passes for your time in the Canadian Rockies national parks. (But your #16 and #5 route will take you through Mt. Robson Provincial Park, and from Clearwater, you can visit Wells Gray PP.)
It can be somewhat confusing as to our parks system in Canada. As mentioned earlier we have national and provincial parks as well as regional and city parks and recreation areas....confusing.
The main two types that you will be most interested in can be found in the following web links.
Lots to see and do...plan as well.
Hopefully that gives you some areas to start your planning, enjoy your visit :0)
According to your intended route above, you will only be in Banff and Jasper. If that takes less than 7 of your 10 days, then you probably only want to buy daily passes. The only proviso to that would be if you wished to visit any of the national historic sites (I think there's one in Calgary, 1 in Banff, and then there are four around Vancouver). The Discovery Pass would also give you access to these sites, so if you intended to visit any of them, then it increases the value of the Discovery Pass to you.
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