Gender Roles and Ideals. In this day and age, no matter if mom or dad is working or taking care of the household chores, children's beliefs about gender roles and ideals still come from parent's beliefs and how they act. Boys can learn from dads values of being a man, and how to treat women. Girls learn from dads how a man should treat her, and what men value in a woman. In fact, one study showed that dad's gender beliefs were more influential on daughters that moms.
Ambition. Both parents can provide ambition in children, but in different ways. For the most part, moms tend to be more cautious about their children than dads. Dads are more likely to encourage children to take risk, both on the playground and in life. An example is when dad encourages kids to climb to the top despite their expressed fear. This can provide children with a bolder sense of ambition that sets a foundation for children's sense of achievement and in making decisions in the future.
Emotional Regulation. Moms get a lot of credit for nurturing and helping children express their feelings. Dads may not be very good at "talking" or "showing" feelings, but they are influential in helping children regulate their emotions. This mainly comes from dad's calm reactions to things from scrapes and bruises to the bullies at school. Dad's ability to let it go or not make a big deal of things, helps children not to dwell on things. Dad's roughhousing with children can also help children learn to control their bodies and emotions. This can allow children to learn how to be in touch with their limits (emotionally and physically) and other's limits.
Dad's involvement in a child's life provides balance. Think of the yin/yang symbol or anything else in life that is symbolic of harmony. Children thrive when there is balance in their life. So to all of the dads out there who are engaged in their children's lives, remember that you play an irreplaceable role. You have an eternal influence on your children and although only one day a year is dedicated to you, just know that you are celebrated in your children's eyes everyday.