Replace capacitors in about half the time
Leave old caps in place, no unsoldering is necessary
No more breaking traces during removal

I've successfully repaired multiple power supply boards by soldering new capacitors in parallel with the bad capacitors. When you put capacitors in parallel you add their values, so if you put a good capacitor in parallel with a bad one, most of the current will go into the new one. The biggest risk to putting them in parallel is that the bad cap may leak and corrode the circuit, but because most of the current will go through the new cap the old cap should deteriorate slowly.

In the first picture the new capacitors are laying on their side, hot melt glued to the front of the board. Jumper wires go from each capacitor to the backside of the board shown in picture 2.

Supplies

Capacitors option 1: Google 'capacitor repair kit BRAND MODEL', there are a bunch on ebay
Capacitors option 2: For individual capacitors Digikey is a good source, get the same or greater voltage rating and same or greater capacitance
Soldering iron
Solder
Flux
Solid core insulated copper wire, cat 5 riser cable works well. DO NOT use patch cables, the wires are stranded and harder to work with.
A good wire stripper will save time, I like this $10 one from Harbor Freight: PITTSBURGH
Self-Adjusting Wire Stripper

Step 1: Make and Solder Jumpers, Hot Glue New Caps in Place

I use insulated solid core copper wire to make jumpers, wires from cat 5 riser cable work well. Stranded wire is harder to work with. For larger capacitors use thicker wire (lower gauge) or put multiple cat 5 strands in parallel to each lead.

Find and mark all the capacitor leads on the back side of the circuit with + and -.

Make jumpers that will go from the back side of the board to the front of the board where the new capacitor will be placed. Keep the jumpers short as possible and twisted together, it will reduce interference.

Strip the ends of the jumpers, solder them to the old capacitor leads and to the new capacitor leads.

Hot melt glue the new capacitor to the top of the board, the jumpers should remain twisted.

Tip1: If a capacitor has long enough leads exposed on the front side of the board, you can cut the capacitor off leaving the old leads and solder the new capacitor to the old leads. This method is even faster. See the last picture for an example.

Tip 2: You should replace all the electrolytic capacitors, not just the visibly bad ones. The other capacitors will likely fail in the future.

PS: You may be wondering how I got so fast at replacing caps? My employer had bought a bunch of flat screen monitors with bad capacitors. When your monitor went out you would get a smaller replacement monitor. ;-( I started fixing everyone's monitors, my charge was a case of beer! I could take apart a monitor, replace all the caps and reassemble in under an hour!

Further reading: Wikipedia has a fascinating article on capacitor plague. https://en.wikipedia.org/wiki/Capacitor_plague