Get a proper pick set from HF, they're short money.

 

A spare small allen key, something like 2mm with the end filed down to a sharp thin point will probably help. Slide the pick long the top of the terminal until you feel it bottom on the back of the connector housing and push up on the tab gently while gently pulling on the wire from behind.
Quite often the pick will jump off of the tab and it's a case of starting again.
It can be difficult the first time you try to get these pins out due to lack of familiarity but once you get the first one, it's pretty simple. Just don't get rough on it from frustration and all will be good.
Remember to record the wire locations in the connector before removing.

 

Pin Extraction - Animated (wait for it)

 

I know your pain. Frustrating as hell. After butchering a few pins, I cut off the wires and went out and got a regular set of connectors. No problems since. You can take every one's advice and good luck but this can drive you to the brink.