"My mother is a doctor."
Translation:Min mamma är läkare.
mamma and pappa are the most common words even among adults. But mor and far sound more formal. Also, those two are used more in the south of Sweden, and sound less formal there than in the rest of the country. moder and fader aren't really used by anyone anymore.
More info here: https://www.duolingo.com/comment/5848211
en doktor can be a person who holds a doctorate in any subject (PhD, Doctor of Technology, Doctor of Medicine, Doctor of Science, whatever), but en läkare is 'a physician', who usually does not have a doctorate degree.
In everyday speech though, we often say doktor when we mean läkare.