The battery is good but the car is like there's no battery at all. No lights no nothing. Won't start, can't jump it. Got the car home on my trailer and later that night my son came home and said the lights are on. Sure enough the parking lights are on. Get inside the car put the key in the ignition and every thing goes black like before.
There are a number of things to check here given the information that you provided, but here are some things that I would do first...
Check battery voltage with a multimeter. While the car is off, it should be around 12.5-12.7 with the car off. If it is lower, it may be enough to power some lights, but not kick on the starter motor.
Check your main fuses/relays. Perhaps the main alternator fuse is blown. Some cars use what is called a 'fusible link' on the alternator which is a thin wire that will burn out before damage is done to the components. Check your manual to see what is used and where it is located.
Check your battery cables. If they're loose, the car's starting can be unpredictable (and dangerous due to sparks!). This may have even triggered the vehicle's antitheft system which would keep you from starting up, but I don't think it would kill everything in the car, so this isn't my first bet. If this is, however, the case, then you'll need to bring it to the stealership.
I don't think this is the issue since the car cannot be jumped, but your alternator itself may be in bad shape as well. If you can get it to a shop, they can run a load test on the battery system and see if your alternator/battery is any good.
I have a very similar problem with my 2005 Kia Spectra5. However, I know the problem. The wire coming off the alternator to the fusible link is damaged. I need to replace it. Depending on the positioning of this wire, I may or may not get ANY power to my car. I'm going to replace this (hopefully today). As a side note, the fusible link is a likely candidate for your issue. I had to replace mine two months ago. I think it blew due to the damage of the wire leading to alternator.