You'll be driving on mountain highways. They aren't dangerous in themselves, not in summer when there won't be snow and ice, but if you don't pay attention, if you go too fast, try to pass someone when you shouldn't, etc etc, there are places where you aren't simply going to roll off the road into a field. You'll hit a rock face, or a semi coming the other way, or end up 500 feet down a canyon in a river. If all the driving you've done so far is around Calgary and the flat parts of Alberta, the mountain roads are probably going to give you a few tense moments. But thousands of people drive that way every day and they make it in one piece.
It's about 550 km from Calgary to Summerland. I always figure on an average speed of 80 kmh when I'm planning a road trip. That allows for faster driving in some places with time allowed for buying gas, getting a coffee, or stopping to walk around and look at stuff, and for getting stuck behind something slow. Let's say it's a 7 hour drive, give or take a bit.
I can understand why your parents might not want you to go, but you sort of have to do stuff like this sooner or later. If you were my kid I'd say do it but for god's sake be careful. Don't exceed the speed limit. If there are signs warning you to slow down for a curve, pay attention to them. Watch the road and the other traffic the whole time. If you want to gawk at the scenery, pull over to do it. Don't talk on your cell phone while you're moving. Make sure your car is in good working order. If you feel stressed, or sleepy, pull over and walk around for 10 minutes. Keep the gas half full or more. In summer there are people all over the place so I don't think your personal safety is much of an issue, no more than it is anywhere else, but lock the car if you get out of it. When I'm travelling alone, I either stop where there are lots of other people or where there's no one else. If someone else shows up and it's not a family or Grama and Grandpa, I leave.
You might want to go Hwy 3 through BC instead of the Trans-Canada. It's a bit more winding, probably a bit slower but usually has less traffic.