You are correct in that the Star Trek helmsman bears a striking resemblance to Jennifer Anniston. However...
The Carol Burnett Show ran from 1967 through 1978:
https://en.wikipedia.org/wiki/The_Carol_Burnett_Show
Jennifer Aniston was born in 1969, meaning that she was only 9 when the show ended:
https://en.wikipedia.org/wiki/Jennifer_Aniston
The Star Trek parody on The Carol Burnett Show actually ran in the first year of The Carol Burnett Show 1967, 2 years before Aniston was born:
https://www.imdb.com/title/tt0536554/plotsummary?ref_=tt_ov_pl
1967 seems about right since at that time, Carol Burnett would be about 34...
https://en.wikipedia.org/wiki/Carol_Burnett
...and Andrea Martin, who plays Mr. Spock, would be about 30:
https://en.wikipedia.org/wiki/Andrea_Martin
What you may be looking at is Jennifer Aniston's MOM, Nancy Dow, who was an actress who was on TV shows such as The Wild Wild West in 1967:
https://www.imdb.com/name/nm0235624/?ref_=nv_sr_1?ref_=nv_sr_1
The imdb.com entry for the Star Trek Parody show on Carol Burnett...
https://www.imdb.com/title/tt0536554/plotsummary?ref_=tt_ov_pl
...only lists the main cast members, so if Nancy Dow were on the show, we have no way of telling. Even Andrea Martin isn't listed either on the Star Trek/Burnett IMDB.com page or on Andera Martin's IMDB page. You can see in the following picture how similar Jennifer and her mother's facial structure was:
https://www.eonline.com/news/994535/jennifer-aniston-details-the-deep-wounds-caused-by-her-late-mother
In addition, check out this photo of a young mother Nancy Dow holding her daughter Jennifer Anniston from the 1970s and see if Nancy Dow might have been the Jennifer look alike:
https://www.pinterest.com/pin/3588874681658095/?lp=true
It might have been Jennifer Aniston's mother, Nancy Dow. It could not possibly have been Aniston herself since she would not be born for 2 years. 
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