Why Dads Don't Babysit
Parenting or raising children by definition includes both parents as equal partners. Both parents are parenting what means spending quality time with their children, take care for their physical and psychological well-being, teach them what is right and what is wrong, etc. For that reason dads don't babysit their own children but they parent, like mothers.
The expression which many dads still like to use probably has its origin in history
when raising children and looking after them was predominantly left over to women.
That does not mean that men were not interested in their children. However, in the
times when all important decisions in family were taken solely by men it did not
even crossed their minds to stay at home so that their wives could spend some time
alone. Men today still like to think they are in charge of everything and to say
that a man is the head of the family. However, things have changed a lot in the
previous century including the parenting style which foresees equal participation of
both parents in raising children, while "trousers" in many families today are worn
by women. Men are well aware of changes that occurred in the roles of the genders in
society and family as well as in parenting. Thus men saying "I have to babysit" when
looking after one's own children usually does not have a pejorative meaning although
it can be also interpreted as "I am forced to do my wife's job while she is away
because she made me to".