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Using Rome2Rio.com, I was able to ask for transportation from Jasper to Edmonton, Calgary, Vancouver and it identifies buses, both Greyhound and Brewsters.
The buses depart from the Jasper Transit Station at 607 Connaught Drive. There is regular Greyhound Bus service to and from Vancouver, BC and Edmonton, AB. If you want to get to Calgary, you transfer in Edmonton.
Jasper to Edmonton, Greyhound 5020, once daily 5 hours 37 minutes leaving at 6:45 am
Jasper to Vancouver Greyhound 5409, once daily 10 h 31 minutes, leaving at 5 am
Jasper to Calgary Brewster once daily 7 hours 30 minutes leaving at 1:45 pm
Parks Canada details how to get to Jasper National Park, located 192 miles west of Edmonton, 256 miles northwest of Calgary and 500 miles northeast of Vancouver.