Yes, it's mother and father.
No, they are not commonly said by children, but rather as more of a regional pronunciation.
In fact, the only time I've ever seen them written that way is in the lyrics to this song.
Children would say "Mommy" and "Daddy" or "Mom" and "Dad." Few children would write a letter to their parents as "Hello Mother, Hello Father." They would write "Dear Mom and Dad."
The song is a parody - and it's VERY funny when you hear it sung.