Like trigley say, don't push it to hard. Your early for the mountains and campgrounds, not just in up the icefeild parkway but just about everywhere north of there.
Third weekend in May is the unofficial opening date of the Canadian camping season. Come May 1st a lot of campgrounds will open, however not all of them will be.
Watch the weather reports, and check road reports, there are a lot of web cams in BC and Alberta, there are not to many in the Yukon.
BC web cams and road reports If you see bad weather coming on your route early in the season or any time, just hold up a day or two and wait it out. Look on the link I gave you ... half way down left side ( DRIVEBC) for road reports.
Also on the web cams for ( southern interior) will show Banff and Lake Louise) yes, they are in Alberta, however BC just wanted to help Alberta advertise the Canadian Rockies. And that's because BC has more of the Canadian Rockies than Alberta does ( grin)
My oldest daughter lived west of Calgary for many years, not to many years ago there was a snow storm in the Banff area on August 1. Next day, you wouldn't know it hit on the road, but all the mountain peaks were still nice and white.
Have a blast and see you up there next summer.
Soup.