Lo again you dirty boy!
Heheh. You have encountered one of the many thrills of MP editing.
In OFP objects of the STATIC(i.e.non-vehicle) class, such as ammo boxes, do not broadcast their position across the network. A parachute, however, is a a vehicle, so its position will be synched across the network.
To solve this problem you must make sure the drop script is run on every machine on the network. To do this I need to be reminded how you are calling the script. Are you using a trigger or an action?
In either case what you want to do is use the trigger or action to broadcast a variable, and then use another trigger that uses that variable to start the drop script.
Assuming a radio trigger. In the trigger OnActivation:
startDropScript = TRUE; publicVariable "startDropScript"
Now add another trigger (none,once) with Condition:
startDropScript
In the On Activation put the call to the drop script.
Some people think that initialising the variable also helps. I do this myself, mostly out of paranoia. If you want to do it too then add the following line to your init.sqs
startDropScript = FALSE