Headlight/brake light issue

The only place headlights and park turn lghts are adjacent is in the forward lamp harness. So if all of these problems began at once, Look for a fault in that harness. Behind the battery isn't as common as at the bulkhead connection but it has happened.
If these lighting problems began at seperate times, isolate and correct one at a time.
A bad headlight switch will turn the lights on and off like you've desicrbed if there is a short, or if its just a bad switch. Modern headlights that draw more current will cause this too. That switch shouldn't effect the turn signals.