Asrock p67 pro 3 error code 62

All the components following a circuit trace should provide resistance though so if you find one that does not that looks anything like these:

They should absolutely have a resistance rating on a test meter. If you find one that does not (it must look visually similar to these, the color can vary) you can try the pencil trick.

For the pencil trick you basically draw a line on the resistor/fuse in pencil that connects the 2 solder points. This should be conductive enough to get signal on the GPU if that is the only damaged component. The thicker the line the greater the conductivity. Generally any component soldered on to the PCB at each end like this should have a resistance rating. Now, once you identify the damaged component/components and tested with the pencil trick you really should replace the broken components with matching ones. There should be numbers on the components that will allow you to order them. If you do get this far you need to be careful to orient the replacements the right way or you will damage other components and probably brick the GPU completely. 

I personally suspect that the outputs are blown. If it was bad memory or a bad core typically the card will display something, usually flashing colors, garbled nonsense etc. There should be tiny fuses connecting to each of the display outputs but also a further array of fuses and circuitry that connects to all of those. If all the outputs are providing no signal then likely something is damaged at a point before the signal is split to all the outputs. Trace the circuits very carefully back from one of the outputs and find the point the signal is split. Examine the components in that area very carefully, you should find a blown fuse, resistor or IC. 

Interestingly, if you know any electronics gurus this problem should be easy for them to trace and correct, if it is only an output issue then it is no different than the circuitry found on a TV, radio or any other device that has inputs/outputs. I have a friend that usually does this kind of stuff for me for free but depending on where you live it may be cheaper and easier to give it to a professional to check the output circuitry.  

Indeed, for that price it is totally worth having some fun with it and trying to make it work again. Once he has done all the things suggested already I can suggest a few more things I have done on old GPUs to breathe life back into them in the past. 

I used to have a bin full of old, dead GPUs that I would pirate parts from (usually fans and coolers) and occasionally when I was bored would try and revive some of them. The most common cause of failures on higher end cards were failed attempts at "unlocking" them via vBIOS updates. In many cases flashing them back to stock revived them but the process can be tricky. The next most common was failed attempts at adding aftermarket cooling/water cooling where the user cracked/chipped the core. Sadly that is not correctable.

I also agree with Parsec and Wardog on the PSU issue, I would replace it and see if your 560 stops artifacting.