Obviously, if it allows you to start the engine in "reverse" (which it shouldn't) and doesn't allow it in "park" or "neutral" (which it should), there is a problem with the start interlock switch not coinciding with what position the shift linkage is actually in.
On that car, I don't know whether that switch is physically located in the console (at your shift lever) or at the transmission. One way or the other, the linkage and the switch, and any adjustments that may be involved, are where you need to focus your investigation.
If "R" is actually Reverse, and "D" is actually Drive, and "N" is actually Neutral, and "P" is actually Park, once the car is running, then the relationship between the shift lever and the transmission is correct, and the problem involves only the start interlock switch. If the actual functions of the transmission are different from what position the lever is in, then the problem is the connection between the shift lever and the transmission itself.