I am in Calgary.
Really, you would be advised not to cycle on the trails up in the mountains...since the weather is unpredictable at this time of year. Calgarians are accustomed to freak snowstorm..even in July. And the city is not even in the mountains/higher elevation.
The ground is still muddy/slippery and soft with melting snow on higher elevation. I know because a work colleague who does go skiing, and hiking, spent last weekend with his girlfriend in Canmore town, not doing a whole lot: not worth hiking in. And he knows Canmore/Kanaskaskis area.
This is not England at all. It is wilderness with animals and bears like to come down eventually too to get more food. We have travelled by car on the Bow Valley Parkway in spring, which should be ok on the roads. But still no, not go cycling on the trails in the parks. But again, the big if, is sudden weather changes...very different from Calgary at times...meaning colder, snowier and icier.
Frankly I don't think it's that pretty at this time of year, the deciduous trees are still abit bare and brown.
Have you gone skiing/snowshoeing or snowboarding this winter in Banff/Kanaskaskis or Canmore area or Lake Louise? Lake Louise got a ton of snow, at least quadruple the amount compared to Calgary. We were there snowshoeing in January.
The parks staff are well trained and know their territory well. Take their advice...
Yes Yoho National Park is gorgeous..Marble Canyon, etc. For a 1-2 day ride around the edges of the parks, etc. on paved roads, would be better. If I was in your shoes, I would wait another 2 wks. or so.