I'm trying to find a movie where this woman is hired to teach a little boy and girl. But the mansion is haunted by the parents of the two kids. The parents can posses the kids I think it takes place around the 17th or 18th century.

The Innocents? They are not possessed by their parents, but the previous governess and her lover.

The story sounds right but it wasn't in black and white. Was it remade. Thank you.

You're welcome. Here's a list of the different versions made:
The Innocents (1961) - Movie connections