I need +12V and around 10A for my project. Each wire on the ATX supply I bought is stranded 20 AWG and is rated for less than 10A. What would happen if I connected multiple +12V wires in parallel to divide the current among the thin wires? The PSU says that it can supply 16A out of the +12V output.
Your PSU seems to have a single-rail 12V output. If you will keep the wires (yellow and black) to all your Molex 4-pin connectors of the same length (or if you plan to re-work the 12V connection to some other connector or directly to a board), you can reasonably assume that the current will be equally distributed, more or less, and you will have all 16A at the end with no problem.
If your PSU has multi-rail topology (has 12V1, 12V2, etc. labelled outputs), you might want to put small serial (equalizing) resistors (0.05 Ohms or less) between ganged outputs from these separate 12V1 and 12V2 etc. outputs, to accommodate possible voltage differences in the rail's outputs.