When the "wrong" item operates, it's nearly always caused by a bad ground on the item you intended to run. The voltage can't get to ground via the normal path so finds a route via another path.

In the case of headlights, there's additional cause for confusion of path taken because
1) low beam lamps also have a high beam filament; both use the same ground terminal on the bulb and
2) all four bulbs are wired to light up during high-beam use. They are all tied together. And nowadays, combination hi/low bulbs are usually in place in all four positions (more avaliable now than the dedicated high or low beam only bulbs).

My recent experience is only with the '86-forward cars. So some of the below could be in error. However:

Driver's front turn signals + running lights ("parking lamps") are grounded at the rear of the battery on fender wall or on acccessory rail by the high/low step relay.

Passenger side front small lights are grounded on passenger fender wall behind the windshield washer tank.

Headlight wiring grounds into the wire harness. If there is a poor ground there, I'd provide a new ground from the headlight connector's lead to an existing good ground screw.

For a graphic showing which bulb terminal is which, see http://www.rallylights.com/useful_info/headlamp_wiring.htm
Remember, that's a view of the BACK of the bulb or connector.
Or test your car wiring carefully with your electrical tester.
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Sven: '89 245, IPD sways, electric rad. fan conversion, 28+ mpg - auto tranny. 850 mi/week commute. '89 245 #2 (wifemobile). '90 244 (spare, runs).