I basically did your route in 1995 starting in Jasper on June 1 and cycling to Browning Montana (at which time we headed west to the coast). We had a great ride that year and followed our original plans with a few deviations. Two passes that we wanted to do were closed, Highwood due to the Elk mating season and Going-to-the-Sun due to the winter snowpack (still 10 ft in the highest areas!!). We used Highway 22 to connect between the TransCanada and Highway 3 - a very nice rolling and winding route in the foothills. The weather was varied - we started off in 25 Celsius temperatures in Jasper but four days into the trip east of Banff we had to hunker down for an extra day and wait out a large rainstorm which caused extensive flooding and damage all over southern Alberta. It turn out that this was one of those "200 year" storms - even the town of Waterton was flooded in spots. When we headed west out of Browning, near East Glacier there had been a snow storm the night before with some slush still on the roads. So expect anything as far as weather.
The hostels on the Parkway are great as are the campgrounds with Lake Louise and Banff having very modern hostel facilities. The highlight for me on this stretch of the trip is Waterton Lakes National Park -- my favourite of the three Alberta rockies National Parks. It is still small and enough off the beaten track not to attract the crowds like Jasper and Banff. The scenery is fantastic for the entire route.
Enjoy your trip.