As previously mentioned, layers will be your friend . . . and no reason to be worried about making a fashion statement. Comfort is paramount.
Here is a
link to a good thread of advice on Trip Advisor.
As for weather expectations, no way to begin to guess. Last year, there was some snow fall in June or July inside Denali National Park. One thing you can count on is rain, regardless of where you are during the spring/summer/fall. Good rain gear is essential, and typically does not include an umbrella. (Locals are rarely seen with them.)
As for Mt. McKinley, you really won't be very close to it at all . . . the closest is the Eielson Visitors Center (approx MM 66 inside Denali NP) and then it will be off in the distance. If you take a flightseeing tour, you could do a glacier landing, which will be chilly but layers should be fine.
I personally do not think you have any reason to lug around a parka. A hoodie or decent fleece should suffice as your final (top) layer under most conditions.