For small children, parents are omnipotent. Kids' parents tell them when to eat, when to go out, when to stay in, and, as in "My Papa's Waltz," when to sleep. Everything in a kid's life is ordained by his or her parents. We see a direct demonstration of the father's power in this poem as he leads his son in a dance. The father even has power over his wife, who stands by watching, helpless to prevent her husband and son from turning the kitchen into a mess.
If the mother has power over her husband, she's choosing not to use it in this poem.
The boy fears his father for his power, but also loves him very much.