There are several national parks around Seattle, you probably want to visit Mount Rainier N.P., and North Cascades N.P..
This is not a very feasible plan, only a little more than 2 weeks, you won't get the chance to really experience the beauty of nature. Banff & Jasper would take a good 10 days, 2 weeks would be better. Olympic National Park would take at least 3 days: one day for each of the 3 ecosystems: beach along Pacific Ocean, glaciers and rainforest.
I would suggest your friend to stay in Canada this time. Take Calgary -> 5 Canadian national parks (Banff, Jasper, Yoho, Kootnay & Glacier) -> Vancouver -> Victoria -> back to Calgary (because one-way car rental is extremely expensive). 2 weeks will be extremely tight for this ambitious schedule, but with careful planning, you can pull this off.