Yup, bad grounds at the bulb sockets will cause a bulb to seek ground through an adjacent bulb which causes both to flash.  Bet I fixed fifty or more early sixty's chevys back in the day when I worked a Sears & Sawbuck.  That's about the only thing that could go sour while the car is setting.

OK, good suggestions. Another question - could a bad ground result in the front signal lights, front parking lights and headlights to stop working simultaneously? Seems strange.

I’m not familiar with 67 wiring but I would imagine that it is not all that much different than the 65-66.  In the 65-66, both headlights have their own ground wire (on the inner fender) and the park/turn signals ground via the bulb connection to its socket (rear tail lights ground in the same manner).  I would think headlight/running light problem may be related to the headlight switch, and suggest checking the light switch connections.  Also, the firewall connector may be the cause, as the headlight, running, directionals all got through this connector … if the same as the 65-66 models.
 
Do check the ground connections, crazy things happen with bad grounds.  Also, I have seen where the turn signal switch can get contaminated with wear particles from the horn ring/button that causes shorts within the turn signal switch.  Suggest pulling the steering wheel and give a look, it will be obvious if these wear particles are present.  If present, clean it out with a small brush and a vacuum.  But note that this should have no impact on the headlights functioning. 

Once again - I am an idiot! So, all front lights (headlights, park lights, turn signals) were inop. Rather than try for the cheapest and easiest fixes first, I decided that it had to be the headlight switch. New switch - no change. Then I decided that it had to be the turn signal switch in the steering column. New switch and much cursing - no change. Then after staring idly at the engine compartment, I found myself looking at lonely black wire hanging under the voltage regulator. Yep, a 10 cent bolt and I'm back in business.  When will I ever learn!

Good point. Actually, I am compiling a large stock of spare parts that I have changed - that did not need changing. 

Show me someone who never made a mistake I'll show you someone who never learned anything. 

Bullet Bob wrote:

TKOPerformance wrote:

Show me someone who never made a mistake I'll show you someone who never learned anything. 

 
Or maybe never DID anything.

Also true.
 

OK, I feel better now. Thanks for the moral support.