I did Rainier late last summer, and It didn't get below frezzing, so I wouldn't worry about a parka. Its pretty much like a backpacking trip, except you need big mounteneering boots and crampons, which you can rent. I used layers topped with a waterproof/windproof shell to keep the wind out.
Tent (If your taking guides they will have them most likely)
Boots (You can rent)
Approach shoes (regular hiking boots)
Crampons (You can rent)
Gaiters
Ice Axe (You can rent)
Harness (depending on your route)
Rope/carabiners/belay devices/pressuik slings/ snow pickets (Guide service will provide these)
Food
Clothing (Layers)
Sleeping bag
Stove (Guide service will have)
Water bottles
Ear plugs, Sunglasses, and Sun Block.
Case of Beer for celebration on top
Here's a starts list. I'm sure i missed some stuff, so others, please add! Hopefully this gives you a good start though.
What guide service are you taking if your getting one? If you don't have all the gear, its almost more expensive to do it by yourself.