PC psu 12v and 5v connection in parallel

No, you can't generally parallel multiple supplies and get higher voltage. Each supply will regulate to a specific voltage. Even if all are rated the same, some will be a little higher and some a little lower. The ones set a little higher will take most of the load. There may also be oscillations as the multiple supplies try to fight each other. In no case does this have anything to do with making more voltage.

It is possible to parallel some supplies to achieve higher overall current. However, the supplies need to be designed or rated for this. Switchers are often easier to design for paralleling. Some even have a current sharing signal. This signal is connected between the supplies, which is used to level the amount of current each supply produces. Some supplies achieve this instead by having a somewhat soft current limit. Even if one supply is set a little higher than the others, it will hit its current limit and its output voltage will drop a little. This causes other supplies to kick in and start producing current to hold up the bus. This is generally not considered as good as deliberate current sharing, but it can work with the right design.

In any case, you're going to need switchers one way or the other. You want 1.4 kW out, so dealing with waste heat is going to be a major design issue. Even at 90% efficiency you still have 156 W of heat to deal with. That's not trivial. You say switchers are too costly, but what you want to do is not trivial and doing it right will cost. If you think switchers are too expensive, take a look at the total cost of the alternatives!

The problem was that I shorted PS_ON to ground without the cables connected to the power supply (it is a modular PSU). The voltage sense wires were not connected to their corresponding partner because the cable was not in place. Hence, the PSU thought the voltage was off and shutdown.

The solution was to look at the pinout, and see which +3.3, +5, +12, and ground pins overlapped on the ATX cable. I shorted those to each other (+3.3 <-> +3.3, +5 <-> +5, etc) and now it works great!