Usually the morning of the race, when the tracks open for "early bird" wagering. Sometimes, for really big races (like the Belmont?) tracks will start taking advance play wagers the day before.
The track handicappers morning line is not relevant to the parimutel odds, but they do provide some clues to bettors. One school of thought suggests betting the overlays - horses whose ML odds were significantly lower than their race-time odds. Many bargains can be had that way, but don't let that be the only factor in determinig which horse to bet. Forget about the price - pick the winner.