Electronic – How to solder under already soldered parts

Actually, you're in luck. Your board uses two kinds of connectors: 1) board-to-board and 2)D-subminiature.

1) The board-to-board connectors. - Using VERY sharp, small wire cutters, cut away the plastic. Be slow and gentle. Use the side of the connector pins to guide the base side of the cutter. With the plastic removed, partially insert the connector pins into a spare female connecter. Just barely engage the ends of your pins to hold them in place, then use a fine-tip soldering iron. Since you can access both sides of the connector, this should be straightforward. The wire cutters you need are specifically call flush wire cutters. Regular old diagonal cutters will not do. Cheap is bad in this application.

2) The D connector - Because the plastic around the pins is elevated, you can do the same sort of surgery as on the board-to-board connectors, and the same sort of soldering. It's harder because you have to go between the outer layer of pins (from the upper set of contacts), but it is possible to get a fine-tip soldering iron which will do the job (Radio Shack and Home Depot are not appropriate sources), and the job is made easier by the fact that, unlike the other connectors, the two rows of PCB pads are staggered. Even if the tip contacts the upper pin leads, the spacing is wide enough that you won't bridge them with solder, and even if you do you can remove any solder traces. You'll also need fine solder for this, .025 dia or less. The trick is to push the soldering iron tip in horizontally while feeding the sold in from above. Steady hands are needed. An Xacto knife is also useful for trimming away the plastic.

I hope you don't have many of these boards to modify.

You might want to try with ChipQuik. It is a solder that has much lower melting point than your everyday solder. When you remove as much solder as you can, flood it with ChipQuik. Given enough solder, it should be possible to keep all points in melted state by reheating them in turn as they turn solid.

Once you do have them all melted at the same time just wiggle the board until you get it out. And don't forget to clean ChipQuik before you resolder things back in.