I believe southerners use ba and northerners use bố for father
That is correct.
I have been taught that ba mẹ = parents at beginning of this lesson. So is it bố mẹ also meant parents ?
Bố = ba same meaning
Bố = ba
Previously má was also accepted for mother?
Yes. There are some regional words that are not included in this course. má is an example (from the southern part of Vietnam).
Can của be omitted here?
Yes. If you omit it, it will still be correct grammatically!
Many Southerners use "má" instead of "mẹ," so you may hear them say "ba má."