I have 67 cutlass. My parking light doesn’t not turn on when the headlights are on. Is this normal for this vehicle or I have an issue. The parking light works fine when the headlights are not on or parking lights setting position is on

 

thats how they are designed, off with the headlights on, my 65 is the same

 

It's the way they were designed. Parking lights only come on with the headlight switch pulled to the first detent.

 

As hard as it is to understand by today's standards, the parking lights did not stay on with the headlamps until '68.

It has annoyed me for decades.

If you want more on this search the site, it has been discussed before.

 

The feds didn't require park lights to be on with the headlights until the 1968 model year. The 1967 and 68 headlight switches are different as a result.