Electrical issues.
Turning the headlights on also turns on parking lights.
That's not how it's supposed to work. With very few exceptions (not including the '64 Dart), pre-1968 cars turn the parking lights on only at the first click of the headlight switch. Pull to the second click and the headlamps come on but the parking lights go off. For '68, one of the new federal safety standards was for lighting, and one of its requirements was that the parking lights stay lit with the headlamps (so other drivers can still see your car's width if a headlight burns out). Making this upgrade deliberately is a good idea and it's not difficult, but that's beside the point at the moment; if your parking lights stay lit when you pull the headlight switch to the second click on a '64 Dart that hasn't been deliberately upgraded to do so, it's a pretty good indication you've got a crossed wire somewhere.
That said, you've got four very likely candidates for the source of the trouble, one at each corner of the car, in the form of the double-filament bulbs used for the park/turn and tail/brake lights. If there is a faulty or inadequate ground at any of these four sockets, current will take unauthorised paths. The parking/tail light circuit includes the dashboard lights, and the turn signal circuit includes the dashboard turn signal pilot lights, so there are easy chances for the kind of mayhem you describe if current gets going the wrong ways along the wrong paths.
Another kind of trouble that can happen at those four locations is that the major (bright) and minor (dim) filaments in the bulb can touch. They're not supposed to, and if they do, that'll bridge two circuits that aren't supposed to be bridged. A variant form of this fault is if someone crams a single-filament bulb where a double-filament bulb is supposed to go. Nominally it won't fit, but it happens often enough, and then the single central contact on the bottom of the bulb can bridge the two side-by-side contacts in the socket.
So check for a good ground at all four corners, and toss in four new bulbs—give the old sockets an assist by getting the bulbs with the bigger crescent-shaped base contacts and the noncorroding nickel-plated base, that's
these or
these (also available locally).
Also, as has been mentioned, you'll want to check for a proper instrument cluster ground.