Burnout Paradise version 1.5 has been released as an automatic update over the PlayStation Network today. This is the update that will bring trophy support to the game.
Unfortunately, Criterion's attempts to get the game to retroactively award trophies have been met with failure. Players will have to start a new game with a new save to earn the trophies. Paradise was originally to have been the first game to retroactively award trophies to players.
Here's what Criterion had to say about all this...
Burnout Paradise saves are locked to your profile. We had hoped to use this to retroactively award trophies, but we discovered during testing that this is not possible. You will therefore require a fresh game-save to receive trophies in Burnout Paradise. If you're familiar with PS3 trophies already, you'll recognise that this works in the same way as all other games that have added trophy support post-release.
Also, it looks like the update might not be live in every territory just yet. Despite showing up as a message in the in-game news board, it's not currently downloading a patch when I fire it up. So your mileage may vary.
Criterion's site has a full list of all the trophies.