91 Corolla won't start

I'm having troubles with my 91 Corolla. Here's what's happening:

It won't start unless it's jumped, in which case, it starts just fine.
After it's been running for about 20 minutes, I power it off, and try to power it back on again, but it does nothing. No lights, no sound, it does absolutely nothing.
I replaced the battery and the alternator and it has the same problems, when I try to turn it on, it does nothing. When I jump start it, everything's fine.
All I can think of is that the wiring is bad. Does anyone else have any suggestions?

with everything off, place a current meter in series with the positive battery terminal, to establish if there is a current draw on you battery with the key off. Also, check all your fuses and relays [if possible]

are any of your better cables showing any excessive battery acid build up? if so...clean it off and make sure it hasn't gotten up into the wire. also make sure the terminals are secured on tightly.