Little is known about Conker's mysterious mom & dad, however they were classed as good parents to him, often letting him play with friends on big adventures and loved and cared for him even when Conker didn't live with them anymore. Every time Conker wanted to hear a bedtime story, his mother would tell him the story of the fabled fairy Panther King.
Conker's mother & father gave him a really good and safe life, they always taught him to be good and not participate in bad stuff, despite Conker's good life, he did not grow up to have a bright future. It is often marked that his parents tried their best to raise him as a good person, but fate would ultimately have Conker grow up into an alcoholic, greedy and uncaring adult.
It is likely they are referred as Mr. & Mrs. Squirrel as it is Conker's last name. It is unknown whether they are both alive or dead.