I'm about to buy Arma 2 through Steam, and I just want to make sure I'll be able to download it on both my dad's and my mom's computers to play. Can I or do I have to pick one?
Yes, you can play your Steam games anywhere you log in. You'll have to download it all over again at each place, but once you do you can play it in both places.
I don't believe Arma 2 uses the Steam Cloud though, so your savegames will not follow you. However, if you are installing Arma 2 to play the popular DayZ mod, your savegame is stored on DayZ's own server and is connected to your Steam account, so that would travel with you just fine.
If you buy a game through Steam you will normally be able to download it and play it on any number of machines. The game's Wikipedia article implies there may have been some limits on the number of downloads, but that's not the case now. You will have to download the game onto each machine you want to play it on.
If you buy the DVD then you should be able to install it on multiple machines as games typically require the disk to be present even though they might not read any data off it during game play.