Yes, you sure can play Sims 3 Island Paradise without the original disc (or a disc at all for that matter!)
There is a catch however:
You'll need the CD key to register with Origin. They will then allow you to download a free copy of the game, no CD required!
The downside to that is it will also take up more space/memory on your computer...
I don't know if it's possible in your country, but I am from korea and many people visit cybercafes and play Sims and other games there for free rather than burden their home machine with installing a full game. Depends how long you spend playing Sim I guess.
Other options would be to look for an illegal torrent download or some other file sharing service. Obviously if it's pirated than that would be illegal, and also there's the risk of viruses... Sorry for the bad news!