2[lean on someone] INFORMAL to put pressure on someone in order to make them do something
The Prime Minister’s been leaning pretty heavily on her to resign.
Synonyms and related words
lean on someone, informal
The song has two separate ideas. The line does not suggest the father did something to the mother. It suggests the father was mean and the mother was strict and the girl got pregnant anyway. The mother "leaned hard" on her to not have sex with boys, but it didn't work. She got pregnant.
The lean hard on someone is the same as to lean heavily on someone.
It is a song about a poor family. "red dirt girl" suggests an unpaved country road where the girl lives. Country lanes are often just dirt roads and some are reddish or orange dirt.
Here is an excellent article about poverty in rural America published in The Guardian in which the author, a woman, states subjects of concern to rural communities very clearly, as it applies to her own situation.
"Even at a midwestern state university, my background –
agricultural work, manual labor, rural poverty, teen pregnancies, domestic chaos, pervasive addiction – seemed like a faraway story to
the people I met."
The article makes clear that teen pregnancy and poverty are typical subjects in poor, rural areas.
poverty in rural Kansas