I rode the Icefields Parkway last summer. We spent 3 nights on the Parkway, all with reservations - June 30 at Rampart Creek Campground, July 1 at Beauty Creek Hostel, July 2,3 at HI Canada hostel in Jasper. I don't remember Rampart Creek as being crowded. Beauty Creek Hostel was manned by extremely bike friendly staff. I would call them and ask if you would be able to camp on their grounds even if no beds are available. The HI Canada hostel in Jasper is extremely nice and quite large. They have a very large lawn area out back. Maybe call them to find out if they would permit tent camping on that lawn if no beds were available? I recognize the dates aren't the same as yours but we were there over a holiday weekend.

Personally, I didn't think jamawani was necessarily referring to you but if there are "no other options," there are no other options to suggest? Since you are starting in Alaska, I think you will find many nights where you have no other options - there are some very remote stretches. We camped in rest areas posted "no camping," we camped in turn outs. We were very bear aware and left the areas cleaner than we found them. We had no problems. I will add that many businesses that once were, were no longer, having closed down (seemingly permanently) due to COVID or had been taken over by construction companies to service their employees only. There was a lot of road construction and laying of fiber optics. I think we had a stretch of 6 days without services so a food carry of 6 days. I recognize this was not a subject of your inquiry but, you might want to confirm in advance that remote services along your route are actually still there.