Lawrence and Vivian were the parents of Stuart Carson. Lawrence was a business associate and friend of Jim Walsh and the two set up Stuart with Jim's daughter, Brenda Walsh, on a blind date, which eventually turned into a relationship. Lawrence was hard on his son, always belittling him and worried he would never amount to anything.
Vivian was mostly just seen on Lawrence's arm or talking with Cindy Walsh and had very little storyline impact.
Eventually, after Stuart and Lawrence spent a whole day arguing, Brenda broke up with him and the Carsons were not seen on the show again.