The distance is manageable. The weather is also something you can work around. Still, jumping in all at once against the advice of your parents while they are still footing some of your bills is tricky. When you're just getting started and money is tight, I would probably look at a used bike before new. It can ease some of the financial strain, and if there's less of a money commitment, perhaps the objections will be lessened. Plus it doesn't sound like you have much experience riding. Talking to a shop guy can be informative, but nothing will tell you more about your riding style and preferences then getting out there and doing it. Starting with a new bike can be an expensive experiment, especially if it ends up being the wrong bike for you.

As for all the other objections, the way I convince people that commuting is a viable option is this: I commute. It sounds like you have an ideal situation right now: You have no car, and are still getting to work and school. If you can't bike in all weather, that still leaves a lot of days that you can bike and do not need a ride. But weather, sweat, and road conditions can all be conquered. I use a combination of rain gear and a change of clothes that I keep at work. I have the advantage of having a bus nearby as well, so if I'm really concerned about the weather, I can use that. If that's an option in your area, that can be your safety net. I use the bus a lot in the morning to save time and to avoid needing to change at work. On the way home, I only use the bus if the weather seems dangerous. A little rain is no big deal and neither is changing clothes when you get home.

So I would say get a cheap bike and try it. There's no argument more convincing to "You can't do that." then "I just did."